# Intermediate Algebra: Functions and Graphs

### 1Linear Models1.1Linear Models1.1.3Problem Set 1.1

#### Warm Up

##### 1.1.3.1.
1. $$\displaystyle 5.5$$
2. $$\displaystyle 12$$
##### 1.1.3.3.
$$I=200+0.09 S$$

#### Skills Practice

##### 1.1.3.5.
$$\dfrac{6}{5}$$
##### 1.1.3.7.
$$y \gt \dfrac{6}{5}$$
##### 1.1.3.9.
$$R = 50- 0.4 w$$

#### Applications

##### 1.1.3.11.
1.  $$t$$ (days) $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ $$h$$ (inches) 6 16 26 36 46
2. $$\displaystyle h=6+2t$$
3. 48 in
4. 33 days
5. $$h=6+2(21)\text{;}$$ $$72=6+2t$$
6. 14 is the initial height of the plants in inches. 1.5 is the number of inches they grow each day.
##### 1.1.3.13.
1.  Altitude (1000 ft) $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ Boiling point ($$\degree$$F) $$212$$ $$210$$ $$208$$ $$206$$ $$204$$ $$202$$
2. $$4\degree$$F
3. Over 4000 feet

### 1.2Graphs and Equations1.2.9Problem Set 1.2

#### Warm Up

1. no
2. yes
3. no
4. no
1. yes
2. no
3. no
4. yes
1. no
2. no
3. yes
4. yes
##### 1.2.9.7.
horizontal: 0.25; vertical: 4
##### 1.2.9.9.
1. horizontal: 5; vertical: 250

#### Skills Practice

##### 1.2.9.11.
$$\dfrac{-15}{4}$$
##### 1.2.9.13.
$$35$$
##### 1.2.9.17.
$$x\ge \dfrac{7}{2}$$
##### 1.2.9.19.
$$y = \dfrac{2}{3} + 24$$
##### 1.2.9.21.
$$y = \dfrac{-7}{13}x + 7$$
##### 1.2.9.23.
$$y = \dfrac{2}{9}x + 17$$

#### Applications

##### 1.2.9.25.
1. $$\displaystyle x = -6$$
2. $$\displaystyle x\lt -6$$
3. $$\displaystyle x\gt -6$$
1. $$\displaystyle x = -6$$
2. $$\displaystyle x\lt -6$$
3. $$\displaystyle x\gt -6$$
##### 1.2.9.27.
1. $$\displaystyle x=0.3$$
2. $$\displaystyle x=0.8$$
3. $$\displaystyle x \le 0.3$$
4. $$\displaystyle x \ge 0.8$$
##### 1.2.9.29.
1. $$\displaystyle x \le 4$$
2. $$\displaystyle x \gt 2$$
##### 1.2.9.31.
1. $$\displaystyle x=4$$
2. $$\displaystyle x\lt 22$$
##### 1.2.9.33.
1. $$\displaystyle p=120+5t$$
2. 141
3. 7.5 min

### 1.3Intercepts1.3.6Problem Set 1.3

#### Warm Up

##### 1.3.6.1.
1. $$\displaystyle 5x$$
2. $$\displaystyle 2y$$
3. 5x+2y=1000
##### 1.3.6.3.
$$y=\dfrac{-3}{5}x+\dfrac{16}{5}$$

#### Skills Practice

##### 1.3.6.5.
1. $$(4,0) \text{,}$$ $$(0,-3)$$
##### 1.3.6.7.
1. $$(-8,0) \text{,}$$ $$(0,5)$$
##### 1.3.6.9.
1. $$\displaystyle (3,0), (0,5)$$
2. $$\displaystyle \left(\dfrac{1}{2},0\right), \left(0,\dfrac{-1}{4}\right)$$
3. $$\displaystyle \left(\dfrac{5}{2},0\right), \left(0,\dfrac{-3}{2}\right)$$
4. $$\displaystyle (p,0), (0,q)$$
5. The value of $$a$$ is the $$x$$-intercept, and the value of $$b$$ is the $$y$$-intercept.
##### 1.3.6.11.
1. $$\displaystyle (0, b)$$
2. $$\displaystyle \left(\dfrac{-b}{m},0\right)$$
##### 1.3.6.13.
$$-2x + 3y = 2400$$
##### 1.3.6.15.
$$3x + 400y = 240$$
##### 1.3.6.17.
1. $$22x+9y=33\text{,}$$ and
2. $$\displaystyle y = \dfrac{11}{3} + \dfrac{-22}{9}x$$
##### 1.3.6.19.
1. $$4x+12y=48\text{,}$$ and
2. $$\displaystyle y = 4 - \dfrac{1}{3}x$$

#### Applications

##### 1.3.6.21.
1. $$\displaystyle 9x$$
2. $$\displaystyle 4y$$
3. $$\displaystyle 9x+4y=1800$$
4. $$(200,0)$$ Delbert must eat 200g of figs daily if he eats no bananas.
5. $$(0,450)$$ Delbert must eat 450g of bananas daily if he eats no figs.
##### 1.3.6.23.
1.  $$~ t ~$$ $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ $$~ h ~$$ $$-400$$ $$-300$$ $$-200$$ $$-100$$ $$0$$
2. $$\displaystyle h=-44+20t$$
3. $$(0,-400) \text{:}$$ The diver starts at a depth of 400 feet. $$(20,0) \text{:}$$ The diver surfaces after 20 minutes.
##### 1.3.6.25.
1. $$$2.40x,$$$$$3.20y$$
2. $$\displaystyle 2.40x + 3.20y = 19,200$$
3. The $$y$$-intercept, 6000 gallons, is the amount of premium that the gas station owner can buy if he buys no regular. The $$x$$-intercept, 8000 gallons, is the amount of regular he can buy if he buys no premium.

### 1.4Slope1.4.6Problem Set 1.4

#### Warm Up

Anthony
Bob’s driveway
##### 1.4.6.5.
$$-1$$
##### 1.4.6.7.
$$\dfrac{-2}{3}$$

#### Skills Practice

##### 1.4.6.9.
1. $$\displaystyle \dfrac{3}{2}$$
2. $$\displaystyle 6,~ \dfrac{3}{2}$$
3. $$\displaystyle -9,~ \dfrac{3}{2}$$
4. 27
##### 1.4.6.11.
$$\dfrac{3}{4}$$
##### 1.4.6.13.
$$-4000$$
##### 1.4.6.15.
1. $$\displaystyle m=\dfrac{-3}{4}$$
14.29 ft
(a)

#### Applications

##### 1.4.6.21.
1. 800
2. $$\displaystyle P=7000+800t$$
##### 1.4.6.23.
1. $$-6$$ liters/day
2. The water supply is decreasing at a rate of 6 liters per day.
3. $$\displaystyle W=48-6t$$
##### 1.4.6.25.
1. Yes, the slope is 0.12
2. 0.12 cm/kg: The spring stretches an extra 0.12 cm for each additional 1 kg mass.
##### 1.4.6.27.
1. The distances to the stations are known.
2. 5.7 km/sec
1. 7.9
2. 8.35 km
3. 2 hrs 5 min

### 1.5Equations of Lines1.5.7Problem Set 1.5

#### Warm Up

##### 1.5.7.1.
 $$x$$ $$-1$$ $$~0~$$ $$~2~$$ $$~3~$$ $$~4~$$ $$y$$ $$10$$ $$8$$ $$4$$ $$2$$ $$0$$
1. 8
2. decreases by 2 units
3. The constant term is the $$y$$-intercept and the coefficient of $$x$$ is the slope.
##### 1.5.7.3.
1. $$\displaystyle \Delta h = 22$$
2. $$\displaystyle \Delta r = -57$$

#### Skills Practice

##### 1.5.7.5.
1. $$\displaystyle y=\dfrac{-3}{2}x+\dfrac{1}{2}$$
2. $$\displaystyle m=\dfrac{-3}{2}; ~b=\dfrac{1}{2}$$
##### 1.5.7.7.
1. $$\displaystyle y=14x-22$$
2. $$\displaystyle m=14; ~b=-22$$
##### 1.5.7.9.
1. $$\displaystyle y=\dfrac{-5}{3}x-6$$
2. $$\displaystyle x=\dfrac{-18}{5}$$
1. II
2. III
3. I
4. IV
##### 1.5.7.13.
1. $$\displaystyle y+5 = -3(x-2)$$
2. $$\displaystyle y=-3x+1$$
##### 1.5.7.15.
$$y=4x+40$$
1. III
2. IV
3. II
4. I

#### Applications

##### 1.5.7.19.
1. $$\displaystyle M = 7000 - 400w$$
2. The slope tells us that Tammy’s bank account is diminishing at a rate of $400 per week, the vertical intercept that she had$7000 (when she lost all sources of income).
##### 1.5.7.21.
$$m = -0.0018$$ degree/foot, so the boiling point drops with altitude at a rate of 0.0018 degree per foot. $$b = 212\text{,}$$ so the boiling point is $$212\degree$$ at sea level (where the elevation $$h = 0$$).
##### 1.5.7.23.
1.  $$h$$ $$8$$ $$20$$ $$C$$ $$645$$ $$1425$$
2. $$\displaystyle C=125+65h$$
3. $$m =65 \text{,}$$ the lesson rate is 65 dollars per hour
##### 1.5.7.25.
1.  $$C$$ $$15$$ $$-5$$ $$F$$ $$59$$ $$23$$
2. $$\displaystyle F=32+\dfrac{9}{5}C$$
3. $$m =\dfrac{9}{5} \text{,}$$ so an increase of $$1\degree$$C is equivalent to an increase of $$\dfrac{9}{5}\degree$$F.
##### 1.5.7.27.
1. $$55\degree$$F
2. 9840 ft
3. $$m=\dfrac{-3}{820}\text{.}$$ The temperature decreases 3 degrees for each increase in altitude of 820 feet.
4. $$(19,133 \frac{1}{3}, 0)\text{.}$$ At an altitude of $$19,133 \frac{1}{3}$$ feet, the temperature is $$0\degree$$F. $$(0,70)\text{.}$$ At an altitude of 0 feet, the temperature is $$70\degree$$F.
##### 1.5.7.29.
1. $$\displaystyle (0,28)$$
2. $$\displaystyle m=\dfrac{1}{2}$$
3. $$\displaystyle y= \dfrac{1}{2}x + 28$$
4. $$\displaystyle 36\degree\text{C}$$

### 1.6Chapter Summary and Review1.6.3Chapter 1 Review Problems

#### 1.6.3.1.

1.  $$n$$ $$100$$ $$500$$ $$800$$ $$1200$$ $$1500$$ $$C$$ $$4000$$ $$12,000$$ $$18,000$$ $$26,000$$ $$32,000$$
2. $$\displaystyle C = 20n + 2000$$

#### 1.6.3.12.

$$\dfrac{-3}{2}$$

#### 1.6.3.13.

$$2$$

#### 1.6.3.14.

$$-0.4$$

#### 1.6.3.15.

$$-1.75$$

neither

both

#### 1.6.3.18.

 $$d$$ $$V$$ $$-5$$ $$-4.8$$ $$-2$$ $$-3$$ $$1$$ $$-1.2$$ $$6$$ $$1.8$$ $$10$$ $$4.2$$

#### 1.6.3.19.

 $$q$$ $$S$$ $$-8$$ $$-8$$ $$-4$$ $$36$$ $$3$$ $$168$$ $$5$$ $$200$$ $$9$$ $$264$$

80 ft

#### 1.6.3.21.

$$m=\dfrac{1}{2} \text{,}$$ $$b= \dfrac{-5}{4}$$

#### 1.6.3.22.

$$m=\dfrac{3}{4} \text{,}$$ $$b= \dfrac{5}{4}$$

#### 1.6.3.23.

$$m=-4 \text{,}$$ $$b= 3$$

#### 1.6.3.24.

$$m=0 \text{,}$$ $$b= 3$$

#### 1.6.3.25.

1. $$\displaystyle y=\dfrac{-2}{3}x +\dfrac{10}{3}$$

#### 1.6.3.26.

1. $$\displaystyle y=\dfrac{3}{2}x -8$$

#### 1.6.3.27.

1. $$\displaystyle T = 62 - 0.0036h$$
2. $$-46\degree\text{F}\text{;}$$ $$108\degree\text{F}$$
3. $$\displaystyle -71\degree\text{F}$$

#### 1.6.3.28.

$$y=\dfrac{-9}{5}x+\dfrac{2}{5}$$

#### 1.6.3.29.

$$y=\dfrac{-5}{2}x+8$$

#### 1.6.3.30.

1.  $$t$$ $$0$$ $$15$$ $$P$$ $$4800$$ $$6780$$
2. $$\displaystyle P = 4800 + 132t$$
3. $$m = 132$$ people/year: the population grew at a rate of $$132$$ people per year.

#### 1.6.3.31.

1. $$m=-2\text{;}$$ $$b=3$$
2. $$\displaystyle y=-2x+3$$

#### 1.6.3.32.

1. $$m=\dfrac{3}{2} \text{;}$$ $$b=-5$$
2. $$\displaystyle y=\dfrac{3}{2}x-5$$

#### 1.6.3.33.

$$\dfrac{3}{5}$$

#### 1.6.3.34.

1. $$(4,0)$$ and $$(0,-6)$$
2. $$\displaystyle \dfrac{3}{2}$$

#### 1.6.3.35.

1. $$\left(\dfrac{8}{3},0 \right)$$ and $$(0,-4)$$
2. $$(4,3) \text{;}$$ No

### 2Applications of Linear Models2.1Linear Regression2.1.5Problem Set 2.1

#### Warm Up

##### 2.1.5.1.
a: II; b: III; c: I; d: IV
##### 2.1.5.2.
a: III; b: IV; c: II; d: I
##### 2.1.5.3.
1.  $$k$$ $$70$$ $$50$$ $$p$$ $$154$$ $$110$$
2. $$\displaystyle p=2.2k$$
3. $$m =2.2$$ pounds/kg is the conversion factor from kilograms to pounds

#### Skills Practice

##### 2.1.5.5.
The slope is $$-2.5\text{,}$$ which indicates that the snack bar sells 2.5 fewer cups of cocoa for each $$1\degree\text{C}$$ increase in temperature. The $$C$$-intercept of 52 indicates that 52 cups of cocoa would be sold at a temperature of $$0\degree\text{C}\text{.}$$ The $$T$$-intercept of 20.8 indicates that no cocoa will be sold at a temperature of $$20.8\degree\text{C}\text{.}$$
##### 2.1.5.9.
1. 74 feet
2. The young whale grows in length about 4.14 foot per month
##### 2.1.5.11.
2 min: $$21\degree$$C; 2 hr: $$729\degree$$C; The estimate at 2 minutes is reasonable; the estimate at 2 hours is not reasonable.

#### Applications

##### 2.1.5.13.
1. 12 seconds
2. 39
3. 11.6 seconds
4. $$\displaystyle y = 8.5 + 0.1x$$
5. 12.7 seconds; 10.18 seconds; The prediction for the 40-year-old is reasonable, but not the prediction for the 12-year-old.
##### 2.1.5.15.
1. 53 sec, 64 sec
2. $$\displaystyle y=5.5x-8.6$$
3. 57.95 sec
##### 2.1.5.17.
1. $$\displaystyle y = 121 + 19.86t$$
2. $$\displaystyle 419$$
##### 2.1.5.19.
1. The graph is above.
2. The slope tells us that the time it takes for a bird to attract a mate decreases by 0.85 days for every additional song it learns.
3. 44.5 days
4. The $$C$$-intercept tells us that a warbler with a repretoire of 53 songs would acquire a mate immediately. The $$B$$-intercept tells us that a warbler with no songs would take 62 days to find a mate. These values make sense in context.

### 2.2Linear Systems2.2.6Problem Set 2.2

18.75

#### Skills Practice

##### 2.2.6.5.
$$(-4,5)$$
##### 2.2.6.7.
$$(-2,3)$$
Inconsistent
Dependent

#### Applications

##### 2.2.6.13.
1. Sporthaus: $$y=500+10x$$
Fitness First: $$y=50+25x$$
2.  $$x$$ Sporthaus Fitness First 6 560 200 12 620 350 18 680 500 24 740 650 30 800 800 36 860 950 42 920 1100 48 980 1250
3. 30 months
##### 2.2.6.15.
1. $$\displaystyle 10x + 5y = 300$$
2. $$\displaystyle y=3x$$
3. $$\displaystyle (12,36)$$
4. She should buy 12 hardbacks and 36 paperbacks.
##### 2.2.6.17.
1. $$\displaystyle S=1.5x$$
2. $$\displaystyle D=120-2.5x$$
3. $$x=30\text{;}$$ $$S=45$$
##### 2.2.6.19.
1. The median age
2. 0.3 years of age per year since 1990
3. See (b) above
4. Slightly less than 14 years since 1990
5. More than half the women are older than the mean age of women.

### 2.3Algebraic Solution of Systems2.3.4Problem Set 2.3

#### Warm Up

##### 2.3.4.1.
$$(4,1)$$

#### Skills Practice

##### 2.3.4.3.
$$(1,2)$$
##### 2.3.4.7.
$$(1,2)$$
Dependent

#### Applications

##### 2.3.4.11.
1. $$\displaystyle S=2.5x;~~D=350-4.5x$$
2. 50 dollars per machine; 125 machines
##### 2.3.4.13.
1.  Pounds % silver Amount of silver First alloy $$x$$ $$0.45$$ $$0.45x$$ Second alloy $$y$$ $$0.60$$ $$0.60y$$ Mixture $$40$$ $$0.48$$ $$0.48(40)$$
2. $$\displaystyle x+y=40$$
3. $$\displaystyle 0.45x+0.60y=19.2$$
4. 32 lb
##### 2.3.4.15.
1.  Rani’s speed in still water: $$x$$ Speed of the current: $$y$$
 Rate Time Distance Downstream $$x+y$$ $$45$$ $$6000$$ Upstream $$x-y$$ $$45$$ $$4800$$
2. $$\displaystyle 45(x+y)=6000$$
3. $$\displaystyle 45(x-y)=4800$$
4. Rani’s speed in still water is 120 meters per minute, and the speed of the current is $$13\dfrac{1}{3}$$ meters per minute.
607.5 mi
##### 2.3.4.19.
1.  Sports coupes Wagons Total Hours of riveting 3 4 120 Hours of welding 4 5 155
2. $$\displaystyle 3x+4y=120$$
3. $$\displaystyle 4x+5y=155$$
4. 20 sports coupes and 15 wagons

### 2.4Gaussian Reduction2.4.6Problem Set 2.4

#### Warm Up

##### 2.4.6.1.
$$(-3,-5)$$
##### 2.4.6.3.
1.  Principal Interest rate Interest Bonds $$x$$ $$0.10$$ $$0.10x$$ Certificate $$y$$ $$0.08$$ $$0.08y$$ Total $$2000$$ —— $$184$$
2. $$\displaystyle x+y=2000$$
3. $$\displaystyle 0.10x+0.08y=184$$

#### Skills Practice

##### 2.4.6.5.
$$(-2,0,3)$$
##### 2.4.6.7.
$$(2,-2,0)$$
##### 2.4.6.9.
$$(2,-3,1)$$
##### 2.4.6.11.
$$\left(\dfrac{1}{2}, \dfrac{2}{3},-3 \right)$$
##### 2.4.6.13.
$$\left(\dfrac{1}{2}, -\dfrac{1}{2},\dfrac{1}{3} \right)$$
Dependent

#### Applications

##### 2.4.6.17.
$$x=40$$ in, $$y=60$$ in, $$z=55$$ in
##### 2.4.6.19.
$$\dfrac{1}{2}$$ lb Colombian, $$\dfrac{1}{4}$$ lb French, $$\dfrac{1}{4}$$ Sumatran

### 2.5Linear Inequalities in Two Variables2.5.5Problem Set 2.5

#### Warm Up

##### 2.5.5.1.
1.  $$x$$ $$-2$$ $$5$$ $$6$$ $$8$$ $$y$$ $$12$$ $$5$$ $$4$$ $$2$$
2. $$\displaystyle x+y=10$$
3. See above.
##### 2.5.5.3.
1. The graph of the equation is a line, and the graph of the inequality is a half-plane. The line is the boundary of the half-plane but is not included in the solution to the inequality.
2. The graph of $$x + y\ge 10,000$$ includes both the line $$x + y=10,000$$ and the half-plane of the corresponding strict inequality.

### 2.6Chapter Summary and Review2.6.3Chapter 2 Review Problems

#### 2.6.3.1.

1. 235 kilojoules
2. $$\displaystyle y=0.106x + 4.6$$
3. $$\displaystyle 156.7\degree\text{C}$$

#### 2.6.3.2.

1. $$45$$ cm
2. $$87$$ cm
3. $$\displaystyle y = 1.2x - 3$$
4. $$69$$ cm
5. $$y = 1.197x - 3.660\text{;}$$ $$68.16$$ cm

6

26 min

#### 2.6.3.5.

$$(-1,2)$$

#### 2.6.3.6.

$$(1.9, -0.8)$$

#### 2.6.3.7.

$$\left(\dfrac{1}{2}, \dfrac{7}{2} \right)$$

#### 2.6.3.8.

$$(1,2)$$

#### 2.6.3.9.

$$(1,2)$$

#### 2.6.3.10.

$$\left(\dfrac{1}{2}, \dfrac{3}{2} \right)$$

#### 2.6.3.11.

Consistent and independent

Inconsistent

Dependent

#### 2.6.3.14.

Consistent and independent

#### 2.6.3.15.

$$(2,0,-1)$$

#### 2.6.3.16.

$$(2,1,-1)$$

#### 2.6.3.17.

$$(2,-5,3)$$

#### 2.6.3.18.

$$\left(2,\dfrac{3}{2},-1\right)$$

#### 2.6.3.19.

$$(-2,1,3)$$

#### 2.6.3.20.

$$(2,-1,0)$$

26

17

#### 2.6.3.23.

$3181.82 at 8%,$1818.18 at 13.5%

#### 4.5.3.20.

1. $$\displaystyle R=1200x-80x^2$$
2. $7.50 3.$4500

#### 4.5.3.21.

1. $$\displaystyle y=60(4+2x)(32-4x)$$
2. 2

#### 4.5.3.22.

1. $$\displaystyle R=(20+x)(500-10x)$$
2. $35 #### 4.5.3.23. Answer. $$a=1,~b=-1,~c=-6$$ #### 4.5.3.24. Answer. $$y=-\dfrac{1}{2}x^2-4x+10$$ #### 4.5.3.25. Answer. 1. $$\displaystyle h=36.98t+5.17$$ 2. 116.1 m, 153.1 m 3. (graph) 4. $$\displaystyle h=-4.858t^2+47.67t+0.89$$ 5. 100.2 m, 113.9 m 6. (graph) 7. quadratic #### 4.5.3.26. Answer. 1. $$\displaystyle y=-0.05x^2-0.003x+234.2$$ 2. $$(-0.03, 234.2)~$$ The velocity of the debris at its maximum height of 234.2 feet. The velocity there is actually zero. #### 4.5.3.27. Answer. $$(-\infty,-2) \cup (3,\infty)$$ #### 4.5.3.28. Answer. $$[-3,4]$$ #### 4.5.3.29. Answer. $$[-1,\dfrac{3}{2}]$$ #### 4.5.3.30. Answer. $$(-\infty,-\dfrac{1}{3}) \cup (2,\infty)$$ #### 4.5.3.31. Answer. $$[-2,2]$$ #### 4.5.3.32. Answer. $$(-\infty,-\sqrt{3}) \cup (\sqrt{3},\infty)$$ #### 4.5.3.33. Answer. 1. $$\displaystyle R=p(220-\dfrac{1}{4}p)$$ 2. $$\displaystyle 400 \lt p \lt 480$$ #### 4.5.3.34. Answer. 1. $$\displaystyle R=p(30-\dfrac{1}{2}p)$$ 2. $$\displaystyle 20 \lt p \lt 40$$ ### 5Functions and Their Graphs5.1Functions5.1.7Problem Set 5.1 #### Warm Up ##### 5.1.7.1. Answer. $$-24$$ ##### 5.1.7.3. Answer. $$\sqrt{20}$$ ##### 5.1.7.5. Answer. $$-3, \dfrac{1}{2}$$ ##### 5.1.7.7. Answer. $$\dfrac{14}{3}$$ #### Skills Practice ##### 5.1.7.9. Answer. 1. input: $$v\text{,}$$ output: $$x$$ 2. 15 3. $$-1, \dfrac{5}{2}\text{.}$$ ##### 5.1.7.11. Answer. 1. $$\displaystyle \dfrac{-1}{3}$$ 2. $$\displaystyle \dfrac{-4}{9}$$ 3. $$\displaystyle \dfrac{-3}{4}$$ 4. $$\displaystyle -0.530$$ #### Applications ##### 5.1.7.13. Answer. (b), (c), (e), and (f) ##### 5.1.7.15. Answer. 1. 60 2. 37.5 3. 20 ##### 5.1.7.17. Answer. 1. $$\displaystyle h(1)=2$$ 2. 1 month 3. 1 month 4. 4000 ##### 5.1.7.19. Answer. 1. Approximately$1920
2. $5 or$15
3. $$\displaystyle f(12) \approx 1920;~~ f(5)=1500,~f(15)=1500$$
4. $$\displaystyle 7\lt d\lt 13$$
##### 5.1.7.21.
1. $$\displaystyle F(1992) = 7.5\%$$
2. $$\displaystyle F(2000) = 4\%$$
3. $$\displaystyle F(1998+ = 4.5\%,~ F(2001) = 4.5\%$$
##### 5.1.7.25.
1. $$N(6000) = 2000\text{:}$$ 2000 cars will be sold at a price of $6000. 2. decrease 3. 30,000. At a price of$30,000, they will sell 400 cars.

### 5.2Graphs of Functions5.2.6Problem Set 5.2

#### Warm Up

##### 5.2.6.5.
1. $$\displaystyle x \gt 0.6$$
2. $$\displaystyle x \lt -0.4$$

#### Applications

##### 5.2.6.11.
1. $$\displaystyle -2, 0, 5$$
2. 2
3. $$\displaystyle h(-2)=0,~h(1)=0,~h(0)=-2$$
4. 5
5. 3
6. increasing: $$(-4,-2)$$ and $$(0,3)\text{;}$$ decreasing: $$(-2,0)$$
##### 5.2.6.13.
1. $$\displaystyle 0, ~\dfrac{1}{2}, ~0$$
2. $$\displaystyle \dfrac{5}{6}$$
3. $$\displaystyle \dfrac{-5}{6},~\dfrac{-1}{6},~\dfrac{7}{6},~\dfrac{11}{6}$$
4. $$\displaystyle 1,~-1$$
5. Max at $$x=-1.5,~0.5\text{,}$$ min at $$x=-0.5,~1.5$$
##### 5.2.6.15.
1. $$f(1000) = 1495\text{:}$$ The speed of sound at a depth of $$1000$$ meters is approximately $$1495$$ meters per second.
2. $$d = 570$$ or $$d = 1070\text{:}$$ The speed of sound is $$1500$$ meters per second at both a depth of $$570$$ meters and a depth of $$1070$$ meters.
3. The slowest speed occurs at a depth of about $$810$$ meters and the speed is about $$1487$$ meters per second, so $$f(810) = 1487\text{.}$$
4. $$f$$ increases from about $$1533$$ to $$1541$$ in the first $$110$$ meters of depth, then drops to about $$1487$$ at $$810$$ meters, then rises again, passing $$1553$$ at a depth of about $$1600$$ meters.
(a) and (d)
##### 5.2.6.19.
1. $$\displaystyle -1, 1$$
2. $$\displaystyle (-1,1)$$
3. $$\displaystyle [-3,-2] \cup [2,3]$$
4. $$\displaystyle [-5,5]$$
##### 5.2.6.21.
1. $$\displaystyle -2,~2$$
2. $$\displaystyle -2.8,~0,~2.8$$
3. $$-2.5 \lt q \lt -1.25$$ and $$1.25 \lt q \lt 2.5$$
4. $$-2 \lt q \lt 0$$ and $$2 \lt q$$
##### 5.2.6.23.
1. $$\displaystyle g(-6)=0,~g(6)=0,~g(0)=6$$
2. none
3. $$g(x)$$ is undefined for those $$x$$-values
##### 5.2.6.25.
1. $$\displaystyle 0,~5$$
2. $$\displaystyle 0,~\dfrac{-3}{2}$$
3. $$\displaystyle \dfrac{5}{6}$$
4. $$\displaystyle -5,~\dfrac{1}{2}$$

### 5.3Some Basic Graphs5.3.4Problem Set 5.3

#### Warm Up

##### 5.3.4.1.
1. $$\displaystyle 4$$
2. $$\displaystyle 2$$
##### 5.3.4.3.
1. $$\displaystyle 2.080$$
2. $$\displaystyle 6.366$$
3. $$\displaystyle -0.126$$
4. $$\displaystyle -1.458$$
##### 5.3.4.5.
1. $$\displaystyle \dfrac{1}{2}$$
2. $$\displaystyle \dfrac{-1}{3}$$
##### 5.3.4.7.
1. $$\displaystyle -9$$
2. $$\displaystyle 9$$
3. $$\displaystyle 9$$
4. $$\displaystyle -4$$
##### 5.3.4.9.
1. $$\displaystyle -50$$
2. $$\displaystyle -43$$
3. $$\displaystyle 144$$

#### Skills Practice

##### 5.3.4.17.
1. $$\displaystyle f$$
2. $$\displaystyle g$$

#### Applications

##### 5.3.4.19.
$$(-\infty,0) \cup [0.5,\infty)$$
##### 5.3.4.21.
(b): 2 units down, (c): 1 unit up
##### 5.3.4.23.
(b): 1.5 units left, (c): 1 unit right
##### 5.3.4.25.
(b): reflected about $$x$$-axis, (c): reflected about $$y$$-axis
1. vi
2. ii
3. iv
4. i
5. v
6. iii
##### 5.3.4.29.
1. horizontal shift of square root $$y=\sqrt{x}$$
2. vertical shift of cube root $$y=\sqrt[3]{x}$$
3. vertical shift of absolute value $$y=\abs{x}$$
4. vertical flip of reciprocal $$y=\dfrac{1}{x}$$
5. vertical flip and vertical shift of cube $$y=x^3$$
6. vertical flip and vertical shift of inverse-square $$y=\dfrac{1}{x^2}$$
##### 5.3.4.31.
1. $$\displaystyle x \lt 2$$
2. $$\displaystyle x \gt \dfrac{-2}{3}$$
##### 5.3.4.33.
1. $$\displaystyle 41$$
2. no solution
3. $$\displaystyle 29\lt x\le 61$$

### 5.4Direct Variation5.4.6Problem Set 5.4

#### Warm Up

##### 5.4.6.1.
1. $$\displaystyle -10$$
2. $$\displaystyle -16, 20$$

#### Skills Practice

##### 5.4.6.3.
1. $$\displaystyle y=0.3x$$
2.  $$x$$ $$2$$ $$5$$ $$8$$ $$12$$ $$15$$ $$y$$ $$-0.6$$ $$1.5$$ $$2.4$$ $$3.6$$ $$4.5$$
3. $$y$$ doubles also
##### 5.4.6.5.
(b), $$k=0.5$$
(c)

#### Applications

##### 5.4.6.9.
1.  $$\text{Price of item}$$ $$18$$ $$28$$ $$12$$ $$\text{Tax}$$ $$1.17$$ $$1.82$$ $$0.78$$ $$\text{Tax}/\text{Price}$$ $$\alert{0.065}$$ $$\alert{0.065}$$ $$\alert{0.065}$$
Yes; $$6.5\%$$
2. $$\displaystyle T = 0.065p$$
##### 5.4.6.11.
1. $$\displaystyle m = 0.165w$$
 $$w$$ $$50$$ $$100$$ $$200$$ $$400$$ $$m$$ $$\alert{8.25}$$ $$\alert{16.5}$$ $$\alert{33}$$ $$\alert{66}$$
2. 19.8 lb
3. 303.03 lb
4. It will double.
##### 5.4.6.13.
1. $$\displaystyle v=15.306 d$$
2. 3985 million light-years
3. 18,979 km/sec
##### 5.4.6.15.
1. $$\displaystyle P = \dfrac{1825}{8192}w^3\approx0.228w^3$$
 $$w$$ $$10$$ $$20$$ $$40$$ $$80$$ $$P$$ $$\alert{223}$$ $$\alert{1782}$$ $$\alert{14,259}$$ $$\alert{114,074}$$
2. 752 kilowatts
3. 33.54 mph
4. It is multiplied by 8.
##### 5.4.6.17.
1. $$\displaystyle d = 0.005v^2$$
2. 50 m
##### 5.4.6.19.
1. $$\displaystyle W = 600d^2$$
2. 864 newtons
##### 5.4.6.20.
2. It is one-ninth as great.
3. It is decreased by 19% because it is 81% of the original.

### 5.5Inverse Variation5.5.4Problem Set 5.5

#### Warm Up

##### 5.5.4.1.
$$R=\dfrac{1}{3}I\text{.}$$ Not inverse variation.
##### 5.5.4.3.
$$W=32,000-d$$ Not inverse variation.

#### Skills Practice

(c)
##### 5.5.4.7.
1. $$\displaystyle y=\dfrac{120}{x}$$
2.  $$x$$ $$4$$ $$8$$ $$20$$ $$30$$ $$40$$ $$y$$ $$30$$ $$15$$ $$6$$ $$4$$ $$3$$
3. $$y$$ is divided by 2
##### 5.5.4.9.
(b), $$k=72$$
(c)

#### Applications

##### 5.5.4.13.
1.  $$\text{Width (feet)}$$ $$2$$ $$2.5$$ $$3$$ $$\text{Length (feet)}$$ $$12$$ $$9.6$$ $$8$$ $$\text{Length}\times \text{width}$$ $$\alert{24}$$ $$\alert{24}$$ $$\alert{24}$$
$$24$$ square feet
2. $$\displaystyle L = \dfrac{24}{w}$$
##### 5.5.4.15.
1. $$\displaystyle B = \dfrac{88}{d}$$
 $$d$$ $$1$$ $$2$$ $$12$$ $$24$$ $$B$$ $$\alert{88}$$ $$\alert{44}$$ $$\alert{7.3}$$ $$\alert{3.7}$$
2. $$8.8$$ milligauss
3. More than $$20.47$$ in
4. It is one half as strong.
##### 5.5.4.17.
1. 645
2. $$\displaystyle D=\dfrac{64,500}{n}$$
3. 215
##### 5.5.4.18.
1. $$\displaystyle m =\dfrac{8}{p}$$
2. 0.8 ton
##### 5.5.4.20.
1. $$\displaystyle T=\dfrac{4000}{d}$$
2. $$8\degree$$C
##### 5.5.4.22.
1. It is one-fourth the original illumination.
2. It is one-ninth the illumination.
3. It is 64% of the illumination.

### 5.6Functions as Models5.6.6Problem Set 5.6

#### Warm Up

##### 5.6.6.1.
1. $$\displaystyle (0, \infty)$$
2. none
3. $$\displaystyle (-infty, 0)$$
4. none
##### 5.6.6.3.
1. none
2. $$\displaystyle (0, \infty)$$
3. none
4. none
##### 5.6.6.5.
1. $$\displaystyle (-infty, 0)$$
2. none
3. $$\displaystyle (0, \infty)$$
4. none
##### 5.6.6.7.
1. $$\displaystyle s=h(t)$$
2. After 3 seconds, the duck is at a height of 7 meters.

#### Skills Practice

1. Increasing
2. Concave up
##### 5.6.6.11.
$$y=\dfrac{k}{x}$$

#### Applications

(b)
(a)
(b)
1. II
2. IV
3. I
4. III
##### 5.6.6.21.
$$y = x^3$$ stretched or compressed vertically
##### 5.6.6.23.
$$y =\dfrac{1}{x}$$ stretched or compressed vertically
##### 5.6.6.25.
$$y =\sqrt{x}$$
##### 5.6.6.27.
1. Table (4), Graph (C)
2. Table (3), Graph (B)
3. Table (1), Graph (D)
4. Table (2), Graph (A)
1. III
2. 3

#### Absolute Value

##### 5.6.6.1.
1. $$\displaystyle \abs{x}=6$$
##### 5.6.6.3.
1. $$\displaystyle \abs{p+3}=5$$
##### 5.6.6.5.
1. $$\displaystyle \abs{t-6}\lt 3$$
##### 5.6.6.7.
1. $$\displaystyle \abs{b+1}\ge 0.5$$
##### 5.6.6.9.
1. $$x = -5$$ or $$x = -1$$
2. $$\displaystyle -7\le x\le 1$$
3. $$x\lt -8$$ or $$x\gt 2$$
##### 5.6.6.11.
1. $$\displaystyle x = 4$$
2. No solution
3. No solution
##### 5.6.6.13.
$$x=\dfrac{-3}{2}$$ or $$x=\dfrac{5}{2}$$
##### 5.6.6.15.
$$q=\dfrac{-7}{3}$$
##### 5.6.6.17.
$$b=-14$$ or $$b=10$$
##### 5.6.6.19.
$$w=\dfrac{13}{2}$$ or $$w=\dfrac{15}{2}$$
No solution
No solution
##### 5.6.6.25.
$$\dfrac{-9}{2}\lt x \lt \dfrac{-3}{2}$$
##### 5.6.6.27.
$$d\le -2~$$ or $$~ d\ge 5$$
All real numbers
##### 5.6.6.31.
$$1.4 \lt t\lt 1.6$$
##### 5.6.6.33.
$$T\le 3.2~$$ or $$~T\ge 3.3$$
No solution

### 5.7Chapter 5 Summary and Review5.7.3Chapter 5 Review Problems

#### 5.7.3.1.

A function: Each $$x$$ has exactly one associated $$y$$-value.

Not a function

#### 5.7.3.3.

Not a function: The IQ of $$98$$ has two possible SAT scores.

A function

#### 5.7.3.5.

$$N(10) = 7000\text{:}$$ Ten days after the new well is opened, the company has pumped a total of $$7000$$ barrels of oil.

#### 5.7.3.6.

$$H(16) = 3\text{:}$$ At 16 mph, the trip takes 3 hours.

#### 5.7.3.7.

$$F(0) = 1, ~~F(-3) =\sqrt{37}$$

#### 5.7.3.8.

$$G(0) = -2, ~~G(20) =\sqrt[3]{12}$$

#### 5.7.3.9.

$$h(8) = -6, ~~h(-8) = -14$$

#### 5.7.3.10.

$$m(5) = 6, ~~m(-40) =-4.8$$

#### 5.7.3.11.

1. $$\displaystyle P(0)=5$$
2. x=5,~x=1

#### 5.7.3.12.

1. R(0)=2
2. $$\displaystyle x=2,~x=-2$$

#### 5.7.3.13.

1. $$\displaystyle f (-2) = 3, ~~f (2) = 5$$
2. $$\displaystyle t = 1, ~~t = 3$$
3. $$t$$-intercepts $$(-3, 0), (4, 0)\text{;}$$ $$f (t)$$-intercept: $$(0, 2)$$
4. Maximum value of $$5$$ occurs at $$t = 2$$

#### 5.7.3.14.

1. $$\displaystyle P(-3)=-2, ~~P(3)=3$$
2. $$\displaystyle z = -5,~ \dfrac{-1}{2},~4$$
3. $$(-4, 0), (-1, 0), (5,0)\text{;}$$ $$(0, 3)$$
4. Maximum value of $$-3$$ occurs at $$z = -2$$

Function

not a function

Not a function

Function

#### 5.7.3.23.

1. $$\displaystyle x = \dfrac{1}{2}= 0.5$$
2. $$\displaystyle x = \dfrac{27}{8}\approx 3.4$$
3. $$\displaystyle x \gt 4.9$$
4. $$\displaystyle x \le 2.0$$

#### 5.7.3.24.

1. $$\displaystyle x = 0.4$$
2. $$\displaystyle x = 3.2$$
3. $$\displaystyle 0 \lt x \le 4.5$$
4. $$x \lt 0$$ or $$x \gt 0.2$$

#### 5.7.3.25.

1. $$\displaystyle x\approx\pm 5.8$$
2. $$\displaystyle x = \pm 0.4$$
3. $$-2.5\lt x \lt 0$$ or $$0\lt x\lt 2.5$$
4. $$x \le -0.5$$ or $$x\ge 0.5$$

#### 5.7.3.26.

1. $$\displaystyle x = 0.5$$
2. $$\displaystyle x = 2.9$$
3. $$\displaystyle 0 \le x \lt 2.3$$
4. $$\displaystyle x \ge 1.7$$

#### 5.7.3.27.

$$y = 1.2x^2$$

#### 5.7.3.28.

$$y = 54x$$

#### 5.7.3.29.

$$y =\dfrac{20}{x}$$

#### 5.7.3.30.

$$y =\dfrac{720}{x^2}$$

#### 5.7.3.31.

1. $$\displaystyle d = 1.75t^2$$
2. 63 cm

#### 5.7.3.32.

1. $$\displaystyle V=\dfrac{4T}{P}$$
2. $$\displaystyle 32$$

#### 5.7.3.33.

$$480$$ bottles

14.0625 lumens

#### 5.7.3.35.

1. $$\displaystyle w = \dfrac{k}{r^2}$$
2. $$3960\sqrt{3}\approx 6860$$ miles

#### 5.7.3.43.

1. $$\displaystyle g(x)=\dfrac{24}{x}$$

#### 5.7.3.44.

1. $$\displaystyle F(x)=-x^3$$

#### 5.7.3.45.

1.  $$x$$ $$0$$ $$4$$ $$8$$ $$14$$ $$16$$ $$22$$ $$y$$ $$24$$ $$20$$ $$16$$ $$10$$ $$8$$ $$2$$
2. $$\displaystyle y = 24 - x$$

#### 5.7.3.46.

1.  $$x$$ $$0$$ $$4$$ $$10$$ $$12$$ $$14$$ $$16$$ $$y$$ $$0$$ $$6$$ $$15$$ $$18$$ $$21$$ $$24$$
2. $$\displaystyle y = \dfrac{3}{2}x$$

#### 5.7.3.47.

1.  $$x$$ $$0$$ $$1$$ $$4$$ $$9$$ $$16$$ $$25$$ $$y$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$
2. $$\displaystyle y = \sqrt{x}$$

#### 5.7.3.48.

1.  $$x$$ $$0.25$$ $$0.50$$ $$1.00$$ $$1.50$$ $$2.00$$ $$4.00$$ $$y$$ $$4.00$$ $$2.00$$ $$1.00$$ $$0.67$$ $$0.50$$ $$0.25$$
2. $$\displaystyle y = \dfrac{1}{x}$$

#### 5.7.3.49.

1.  $$x$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$y$$ $$5$$ $$0$$ $$-3$$ $$-4$$ $$-3$$ $$0$$
2. $$\displaystyle y = x^2-4$$

#### 5.7.3.50.

1.  $$x$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$4$$ $$y$$ $$0$$ $$5$$ $$8$$ $$9$$ $$8$$ $$-7$$
2. $$\displaystyle y = 9 - x^2$$

### 6Powers and Roots6.1Integer Exponents6.1.5Problem Set 6.1

#### Warm Up

##### 6.1.5.1.
1. $$\displaystyle -2z^2$$
2. $$\displaystyle -24z^4$$
##### 6.1.5.3.
1. $$\displaystyle -12x^3y^4$$
2. $$\displaystyle \dfrac{1}{4ab^4}$$
##### 6.1.5.5.
1. $$\displaystyle 4x^{10}y^{14}$$
2. $$\displaystyle x^4y$$

#### Skills Practice

##### 6.1.5.7.
1. $$\displaystyle 9$$
2. $$\displaystyle \dfrac{1}{9}$$
3. $$\displaystyle 9$$
4. $$\displaystyle \dfrac{1}{9}$$
5. $$\displaystyle -9$$
6. $$\displaystyle \dfrac{-1}{9}$$
##### 6.1.5.9.
1. $$\displaystyle 40$$
2. $$\displaystyle \dfrac{5}{8}$$
3. $$\displaystyle \dfrac{5}{8}$$
4. $$\displaystyle \dfrac{1}{40}$$
5. $$\displaystyle 40$$
6. $$\displaystyle \dfrac{8}{5}$$
##### 6.1.5.11.
1. $$\displaystyle \dfrac{3}{4}$$
2. $$\displaystyle \dfrac{1}{8}$$
3. $$\displaystyle \dfrac{1}{2}$$
4. $$\displaystyle \dfrac{9}{4}$$
5. $$\displaystyle \dfrac{1}{6}$$
6. $$\displaystyle 2$$
7. $$\displaystyle \dfrac{4}{3}$$
8. $$\displaystyle 8$$
##### 6.1.5.13.
1. $$\displaystyle \dfrac{1}{(m-n)^2}$$
2. $$\displaystyle \dfrac{1}{y^2}+\dfrac{1}{y^3}$$
3. $$\displaystyle \dfrac{2p}{q^4}$$
4. $$\displaystyle \dfrac{-5x^5}{y^2}$$
##### 6.1.5.15.
$$1.25$$
##### 6.1.5.17.
$$0.2$$
##### 6.1.5.19.
1. $$\displaystyle \dfrac{20}{x^3}$$
2. $$\displaystyle \dfrac{1}{3u^{12}}$$
3. $$\displaystyle 5^8t$$
##### 6.1.5.21.
1. $$\displaystyle \dfrac{1}{3}x+3x^{-1}$$
2. $$\displaystyle \dfrac{1}{4}x^{-2}-\dfrac{3}{2}x^{-1}$$
##### 6.1.5.23.
$$x - 3 + 2x^{-1}$$
##### 6.1.5.25.
$$-4 - 2u^{-1} + 6u^{-2}$$
##### 6.1.5.27.
$$4x^{-2}(x^4 + 4)$$
##### 6.1.5.29.
1. $$\displaystyle 2.85 \times 10^2$$
2. $$\displaystyle 8.372 \times 10^6$$
3. $$\displaystyle 2.4 \times 10^{-2}$$
4. $$\displaystyle 5.23 \times 10^{-4}$$

#### Applications

##### 6.1.5.31.
1.  $$x$$ $$1$$ $$2$$ $$4.5$$ $$6.2$$ $$9.3$$ $$g(x)$$ $$1$$ $$0.125$$ $$0.011$$ $$0.0042$$ $$0.0012$$
2. they decrease
3.  $$x$$ $$1.5$$ $$0.6$$ $$0.1$$ $$0.03$$ $$0.002$$ $$f(x)$$ $$0.30$$ $$4.63$$ $$1000$$ $$37,037$$ $$125 \times 10^6$$
4. they increase
##### 6.1.5.33.
1. $$\displaystyle P = 0.355v^3$$
2. $$v\approx 52.03$$ mph
3. 3.375
##### 6.1.5.35.
1. $$\displaystyle d=50f^{-1}$$
2. The are of the aperture decreases by a factor of 0.5 at each $$f$$-stop.
##### 6.1.5.36.
1. $$\displaystyle 1.905 \times 10^{13}$$

#### 6.6.3.27.

1. It is the cost of producing the first ship.
2. $$C = \dfrac{12}{ \sqrt[8]{x}}$$ million
3. About $11 million; about 8.3% ; about 8.3%> 4. About 8.3% #### 6.6.3.29. Answer. 1. $$\displaystyle 4096$$ 2. $$\displaystyle \dfrac{1}{8}$$ 3. $$\displaystyle 36\sqrt{3} \approx 62.35$$ 4. $$\displaystyle 400,000$$ #### 6.6.3.30. Answer. 1. $$\displaystyle -27$$ 2. $$\displaystyle \dfrac{-3}{4}$$ 3. $$\displaystyle -3\sqrt[3]{400} \approx -22.1$$ 4. $$\displaystyle -300$$ #### 6.6.3.31. Answer. $$169$$ #### 6.6.3.32. Answer. $$49$$ #### 6.6.3.33. Answer. $$16$$ #### 6.6.3.34. Answer. $$-16$$ #### 6.6.3.35. Answer. $$1,~4$$ #### 6.6.3.36. Answer. $$8$$ #### 6.6.3.37. Answer. $$9$$ #### 6.6.3.38. Answer. $$12$$ #### 6.6.3.39. Answer. $$7$$ #### 6.6.3.40. Answer. $$\pm 8$$ #### 6.6.3.41. Answer. $$5$$ #### 6.6.3.42. Answer. $$4$$ #### 6.6.3.43. Answer. $$g=\dfrac{2v}{t^2}$$ #### 6.6.3.44. Answer. $$r=\dfrac{\pm \sqrt{3q^2-6q+7}}{2}$$ #### 6.6.3.45. Answer. $$p=\pm 2 \sqrt{R^2-R}$$ #### 6.6.3.46. Answer. $$r=\pm \sqrt{2q^3-1}$$ #### 6.6.3.47. Answer. 1. $$\displaystyle \dfrac{5p^4}{a^2}\sqrt{5p}$$ 2. $$\displaystyle \dfrac{2}{w^2}\sqrt[3]{3v^2}$$ #### 6.6.3.48. Answer. 1. $$\displaystyle a^2b$$ 2. $$\displaystyle xy$$ #### 6.6.3.49. Answer. 1. $$\displaystyle 2\sqrt[3]{a^3-2b^6}$$ 2. $$\displaystyle -4ab^2\sqrt[3]{2}$$ #### 6.6.3.50. Answer. 1. $$\displaystyle 2t\sqrt{1+6t^4}$$ 2. $$\displaystyle 4t^4\sqrt{6}$$ #### 6.6.3.51. Answer. 1. $$\displaystyle x^2-4x\sqrt{x}+4x$$ 2. $$\displaystyle x^2-4x$$ #### 6.6.3.52. Answer. 1. $$\displaystyle 14-4\sqrt{6}$$ 2. $$\displaystyle 2a-4b$$ #### 6.6.3.53. Answer. 1. $$\displaystyle \dfrac{7\sqrt{5y}}{5y}$$ 2. $$\displaystyle 3\sqrt{2d}$$ #### 6.6.3.54. Answer. 1. $$\displaystyle \dfrac{\sqrt{33rs}}{11s}$$ 2. $$\displaystyle \dfrac{\sqrt{13m}}{m}$$ #### 6.6.3.55. Answer. 1. $$\displaystyle \dfrac{-3\sqrt{a}+6}{a-4}$$ 2. $$\displaystyle \dfrac{-3\sqrt{z}-12}{z-16}$$ #### 6.6.3.56. Answer. 1. $$\displaystyle \dfrac{2x^2+x\sqrt{3}-3}{x^2-3}$$ 2. $$\displaystyle \dfrac{5m^2-7m\sqrt{3}+6}{25m^2-12}$$ ### 7Exponential Functions7.1Exponential Growth and Decay7.1.7Problem Set 7.1 #### Warm Up ##### 7.1.7.1. Answer. 1.$28
2. #31.36
3. No. It increase by 12% of different amounts.
Decreased by 1%.
4
##### 7.1.7.7.
$$\pm 1.2$$
##### 7.1.7.9.
$$-2.14\text{;}$$ $$0.14$$

#### Skills Practice

##### 7.1.7.11.
1. $$P = 1200 + 150t\text{;}$$ 1650
2. $$P = 1200\cdot 1.5^t\text{;}$$ 4050
##### 7.1.7.13.
1. $$V = 18,000 - 2000t\text{;}$$ $8000 2. $$V = 18,000\cdot 0.8^t\text{;}$$$5898.24
##### 7.1.7.15.
20%, 2%, 7.5%, 100%, 115%
##### 7.1.7.17.
1. $$\displaystyle P_0 = 4,~ b = 2^{1/3}$$
2. $$\displaystyle P(t) = 4\cdot 2^{t/3}$$
##### 7.1.7.18.
1. Initial value $$80\text{,}$$ decay factor $$\frac{1}{2}$$
2. $$\displaystyle f(x) = 80\cdot \left(\dfrac{1}{2} \right)^x$$
##### 7.1.7.19.
The growth factor is $$1.2\text{.}$$
 $$x$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$Q$$ $$20$$ $$24$$ $$28.8$$ $$34.56$$ $$41.47$$
##### 7.1.7.20.
The decay factor is $$0.8\text{.}$$
 $$w$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$N$$ $$120$$ $$96$$ $$76.8$$ $$61.44$$ $$49.15$$
##### 7.1.7.21.
The decay factor is $$0.8\text{.}$$
 $$t$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$C$$ $$10$$ $$8$$ $$6.4$$ $$5.12$$ $$4.10$$
##### 7.1.7.22.
The growth factor is $$1.1\text{.}$$
 $$n$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$B$$ $$200$$ $$220$$ $$242$$ $$266.2$$ $$292.82$$

#### Applications

##### 7.1.7.23.
1.  Years after 2010 $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ Windsurfers $$1500$$ $$1680$$ $$1882$$ $$2107$$ $$2360$$
2. $$\displaystyle S(t)=1200(1.12)^t$$
3. 2644; 5844
##### 7.1.7.24.
1.  Years after 1983 $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ Value of house $$20,000$$ $$25,526$$ $$32,578$$ $$41,579$$ $$53,066$$
2. $$\displaystyle V(t)=200,000(1.05)^t$$
3. $359,171.27;$458,403.66
##### 7.1.7.25.
1.  Weeks $$0$$ $$6$$ $$12$$ $$18$$ $$24$$ Bees $$2000$$ $$5000$$ $$12,500$$ $$31,250$$ $$78,125$$
2. $$\displaystyle P(t)=2000(2.5)^{t/6}$$
##### 7.1.7.27.
1.  Weeks $$0$$ $$2$$ $$4$$ $$6$$ $$8$$ Mosquitos $$250,000$$ $$187,500$$ $$140,625$$ $$105,469$$ $$79,102$$
2. $$\displaystyle P(t)=250,000(0.75)^{t/2}$$
3. 162,280; 68,504
##### 7.1.7.29.
1.  Years $$0$$ $$3$$ $$6$$ $$9$$ $$12$$ Value of boat $$15,000$$ $$13,500$$ $$12,150$$ $$10,935$$ $$9841.50$$
2. $$\displaystyle V(t)=15,000(0.885)^t$$
3. $4995.52;$4421.04
##### 7.1.7.31.
1. $$\displaystyle P(t)=1,545,387 b^t$$
2. $$\displaystyle b=1.049;~ r=4.9\%$$
3. 3,167,157
##### 7.1.7.33.
1. 365
2. $$\displaystyle N(t)=365(0.356)^t$$
3. 0.03
##### 7.1.7.35.
2. 0.765; 23.5%
3. 3080; 4656
1. 39; 1.045
2. 36; 1.047
3. Species B

### 7.2Exponential Functions7.2.7Problem Set 7.2

#### Warm Up

##### 7.2.7.1.
1. $$\displaystyle 3^{x+4}$$
2. $$\displaystyle 3^{4x}$$
3. $$\displaystyle 12^x$$
##### 7.2.7.3.
1. $$\displaystyle b^{-2t}$$
2. $$\displaystyle b^{t/2}$$
3. $$\displaystyle 1$$
##### 7.2.7.5.
$$0.06$$

#### Skills Practice

##### 7.2.7.7.
$$\dfrac{2}{3}$$
##### 7.2.7.8.
$$\dfrac{-1}{4}$$
##### 7.2.7.9.
$$\dfrac{1}{7}$$
##### 7.2.7.11.
$$\dfrac{-5}{4}$$
##### 7.2.7.13.
$$\pm 2$$
##### 7.2.7.15.
$$2.26$$
##### 7.2.7.17.
1. $$(0,26)\text{;}$$ increasing
2. $$(0,1.2)\text{;}$$ decreasing
3. $$(0,75)\text{;}$$ decreasing
4. $$(0,\frac{2}{3}) \text{;}$$ increasing
1. Power
2. Exponential
3. Power
4. Neither
##### 7.2.7.19.
 $$x$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$f(x)=3^x$$ $$\frac{1}{27}$$ $$\frac{1}{9}$$ $$\frac{1}{3}$$ $$1$$ $$3$$ $$9$$ $$27$$ $$g(x)=\left(\frac{1}{3} \right)^x$$ $$27$$ $$9$$ $$3$$ $$1$$ $$\frac{1}{3}$$ $$\frac{1}{9}$$ $$\frac{1}{27}$$
##### 7.2.7.21.
Because they are defined by equivalent expressions, (b), (c), and (d) have identical graphs.
##### 7.2.7.23.
$$0.17; 5.95$$
##### 7.2.7.25.
1. $$\displaystyle P_0 = 800$$
2.  $$x$$ $$0$$ $$1$$ $$2$$ $$g(x)$$ $$800$$ $$200$$ $$50$$
3. $$\displaystyle a = \dfrac{1}{4}$$
4. $$\displaystyle g(x) = 800(\dfrac{1}{4})^x$$
##### 7.2.7.27.
1. Power $$y=100 x^{-1}$$
2. Exponential $$P=\frac{1}{4} \cdot 2^x$$

#### Applications

##### 7.2.7.28.
1. $$\displaystyle V(t) = 20,000(0.8)^{t/3}$$
2. 6 years
##### 7.2.7.29.
1. $$\displaystyle N(t) = 26(2)^{t/6}$$
2. 72 days later
##### 7.2.7.30.
1. $$\displaystyle F_0=400$$
2. $$\displaystyle b=1.06$$
3. $$\displaystyle F(p) = 440(1.06)^p$$
##### 7.2.7.31.
1. $$\displaystyle S_0=150$$
2. $$\displaystyle b\approx 0.55$$
3. $$\displaystyle S(d) = 150(0.55)^d$$
##### 7.2.7.32.
 $$x$$ $$f(x)=x^2$$ $$g(x)=2^x$$ $$-2$$ $$4$$ $$\frac{1}{4}$$ $$-1$$ $$1$$ $$\frac{1}{2}$$ $$0$$ $$0$$ 1 $$1$$ $$1$$ $$2$$ $$2$$ $$4$$ $$4$$ $$3$$ $$9$$ $$8$$ $$4$$ $$16$$ $$16$$ $$5$$ $$25$$ $$32$$
1. $$\displaystyle Three$$
2. $$\displaystyle x = -0.77,~2,~4$$
3. $$\displaystyle (-0.77, 2) \cup (4,\infty)$$
4. $$\displaystyle g(x)$$

### 7.3Logarithms7.3.7Problem Set 7.3

#### Warm Up

##### 7.3.7.1.
1. $$\displaystyle P(t)=300(1.15)^t$$
2. $$\displaystyle 500=300(1.15)^t$$
3. $$\displaystyle t \approx 3.65$$

#### Skills Practice

##### 7.3.7.3.
1. $$\displaystyle 2$$
2. $$\displaystyle 4$$
##### 7.3.7.5.
1. $$\displaystyle \dfrac{1}{2}$$
2. $$\displaystyle -1$$
##### 7.3.7.7.
1. $$\displaystyle -1$$
2. $$\displaystyle 4$$
##### 7.3.7.9.
1. $$\displaystyle 3 \lt x \lt 4$$
2. $$\displaystyle -1 \lt y \lt 0$$
##### 7.3.7.11.
$$-0.23$$
##### 7.3.7.13.
$$0.77$$
##### 7.3.7.15.
$$2.53$$
##### 7.3.7.17.
1. $$\displaystyle \log_t {16} = \dfrac{3}{2}$$
2. $$\displaystyle \log_{0.8}{M} = 1.2$$
##### 7.3.7.19.
1. $$\displaystyle 16^w = 256$$
2. b^{-2} = 9
##### 7.3.7.21.
1. $$\displaystyle x=\log_4{2.5} \approx 2.7$$
2. $$\displaystyle x=\log_2{0.2} \approx -2.3$$
##### 7.3.7.23.
$$2.77$$

#### Applications

1. 2030
2. 6.2%
1. 9.60 in
2. 3.85 mi
1. 2,018,436
2. 5.17%
3. 2008
##### 7.3.7.31.
1. $$\displaystyle 4$$
2. $$\displaystyle 4$$
3. $$\displaystyle 64$$
4. $$\displaystyle \sqrt[3]{16} \approx 2.52$$

### 7.4Properties of Logarithms7.4.5Problem Set 7.4

#### Warm Up

##### 7.4.5.1.
1. $$\displaystyle \log_8 {\dfrac{1}{2}} = \dfrac{-1}{3}$$
2. $$\displaystyle \log_5 {46} = x$$
##### 7.4.5.3.
1. $$\displaystyle 10^8$$
2. $$2;~6;~8~$$ Property (1)
##### 7.4.5.5.
1. $$\displaystyle b^3$$
2. $$8;~5;~3~$$ Property (2)
##### 7.4.5.7.
1. $$\displaystyle 10^{15}$$
2. $$15;~3~$$ Property (3)
##### 7.4.5.9.
1. $$\displaystyle 5$$
2. $$\displaystyle 6$$
3. $$\displaystyle 5$$
(a) and (c) are equal.
##### 7.4.5.11.
1. $$\displaystyle \log 24\approx 1.38$$
2. $$\displaystyle \log 240\approx 2.38$$
3. $$\displaystyle \log 230\approx 2.36$$
None are equal.

#### Skills Practice

##### 7.4.5.13.
1. $$\displaystyle \log_t {16} = \dfrac{3}{2}$$
2. $$\displaystyle \log_{0.8}{M} = 1.2$$
3. $$\displaystyle \log_{3.7}{Q} = 2.5$$
4. $$\displaystyle \log_3 {2N_0} = -0.2t$$
##### 7.4.5.15.
1. $$\displaystyle x=\log_4{2.5} \approx 2.7$$
2. $$\displaystyle x=\log_2{0.2} \approx -2.3$$
##### 7.4.5.17.
1. $$\displaystyle \log_b 4$$
2. $$\displaystyle \log_4(x^2y^3)$$
##### 7.4.5.19.
1. $$\displaystyle \log 2x^{5/2}$$
2. $$\displaystyle \log (t-4)$$
##### 7.4.5.21.
$$y = \dfrac{1}{25}$$
##### 7.4.5.23.
$$b = 100$$
##### 7.4.5.25.
$$2.81$$
##### 7.4.5.27.
$$-1.61$$
##### 7.4.5.29.
$$-12.49$$
##### 7.4.5.31.
1. $$\displaystyle 1.7918$$
2. $$\displaystyle -0.9163$$
(b) and (c)

#### Applications

##### 7.4.5.37.
1. $$\displaystyle S (t) = S_0(1.09)^t$$
2. $$4.7$$ hours
##### 7.4.5.39.
1. $$\displaystyle C(t) = 0.7(0.80)^t$$
2. After 2.5 hours
##### 7.4.5.41.
1. $$\displaystyle S(t) = S_0 \cdot 0.9527^t$$
2. 28.61 hours
##### 7.4.5.43.
$$k=\dfrac{\log\left(\dfrac{N}{N_0}\right)}{t \log {a}}$$
##### 7.4.5.45.
$$t=\dfrac{1}{k}\log\left(\dfrac{A}{A_0}+1\right)$$

### 7.5Exponential Models7.5.4Problem Set 7.5

#### Warm Up

4.16
16
##### 7.5.4.5.
$$y=\dfrac{-2}{3}x-1$$
##### 7.5.4.7.
$$y=\dfrac{-1}{2}x+4$$

#### Skills Practice

##### 7.5.4.9.
$$P(x)=8(0.5)^x$$
##### 7.5.4.11.
$$y=1.5(3){x/5}$$
##### 7.5.4.13.
1. $$\displaystyle y=2.6-1.3x$$
2. $$\displaystyle y=2.5(0,5)^x$$

#### Applications

##### 7.5.4.15.
$$P(t)=2000(2^{t/5};~$$ 14.9%
##### 7.5.4.17.
$$D(t)=D_0(0,5^{t/18};~$$ 3.8%
##### 7.5.4.19.
1. $$\displaystyle P = P_0 (2)^{t/25 }$$
2. 2.81%
##### 7.5.4.21.
1. $$\dfrac{\log 0.5}{\log 0.946}\approx 12.5$$ hours
2. 25 hours
##### 7.5.4.23.
1. $$\displaystyle D(t)=D_0(0.5)^{t/15}$$
2. After 89.5 years, or in 2060
##### 7.5.4.25.
1. $$\displaystyle A = A_0 \left(\dfrac{1}{2} \right)^{t/5730}$$
12.9%

### 7.6Chapter 7 Summary and Review7.6.3Chapter 7 Review Problems

#### 7.6.3.1.

1. $$\displaystyle D = 8(1.5)^{t/5}$$
2. $$18\text{;}$$ $$44$$

#### 7.6.3.3.

1. $$\displaystyle M = 100(0.85)^t$$
2. $$52.2$$ mg; $$19.7$$ mg

#### 7.6.3.9.

$$\dfrac{-4}{3}$$

#### 7.6.3.11.

$$-11$$

#### 7.6.3.13.

$$4$$

#### 7.6.3.15.

$$-1$$

#### 7.6.3.17.

$$-3$$

#### 7.6.3.19.

$$2^{x-2} = 3$$

#### 7.6.3.20.

$$n^{p-1} = q$$

#### 7.6.3.21.

$$\log_{0.3}(x + 1) = -2$$

#### 7.6.3.23.

$$-1$$

#### 7.6.3.25.

$$4$$

#### 7.6.3.27.

$$\dfrac{\log 5.1}{1.3}\approx 0.5433$$

#### 7.6.3.29.

$$\dfrac{\log (2.9/3)}{-0.7}\approx 0.21$$

0.054

2.959

0.195

2.823

#### 7.6.3.35.

$$\log_b x + \dfrac{1}{3} \log_b y - 2 \log_b z$$

#### 7.6.3.37.

$$\dfrac{4}{3} \log x -\dfrac{1}{3} \log y$$

#### 7.6.3.39.

$$\log\sqrt[3] {\dfrac{x}{y^{2}}}$$

#### 7.6.3.41.

$$\log {\dfrac{1}{8}}$$

#### 7.6.3.43.

$$\dfrac{\log 63}{\log 3}\approx 3.77$$

#### 7.6.3.45.

$$\dfrac{\log 50}{-0.3\log 6}\approx -7.278$$

#### 7.6.3.47.

$$\dfrac{\log(N/N_0)}{k}$$

#### 7.6.3.49.

1. $$\displaystyle 238$$
2. $$\displaystyle 2010$$

#### 7.6.3.51.

1. $$\displaystyle C = 90(1.06)^t$$
2. $$$94.48$$ 3. $$5$$ years ### 8Polynomial and Rational Functions8.1Polynomial Functions8.1.6Problem Set 8.1 #### Warm Up ##### 8.1.6.1. Answer. 1. $$\displaystyle a^2 + 2ab + b^2$$ 2. $$\displaystyle a^2 - 2ab + b^2$$ ##### 8.1.6.3. Answer. 1. $$\displaystyle (x-7)^2$$ 2. cannot be factored 3. $$\displaystyle (x+3)^2$$ 4. $$\displaystyle (x+8)(x-8)$$ #### Skills Practice ##### 8.1.6.5. Answer. (b) and (c) are not polynomials; they have variables in a denominator. ##### 8.1.6.7. Answer. 1. $$\displaystyle -1.9x^3+x+6.4$$ 2. $$\displaystyle -2x^2+6xy+2y^3$$ ##### 8.1.6.9. Answer. 1. 4 2. 5 3. 7 ##### 8.1.6.11. Answer. 1. $$\displaystyle 6a^4-5a^3-5a^2+5a-1$$ 2. $$\displaystyle y^4+5y^3-20y-16$$ ##### 8.1.6.13. Answer. 1. $$\displaystyle 1+15\sqrt{t}+75t+125t\sqrt{t}$$ 2. $$\displaystyle 1-\dfrac{9}{a}+\dfrac{27}{a^2}-\dfrac{27}{a^3}$$ ##### 8.1.6.15. Answer. 1. $$\displaystyle 27a^3-8b^3$$ 2. $$\displaystyle 8a^3 + 27b^3$$ ##### 8.1.6.17. Answer. $$(a-2b)(a^2+2ab+4b^2)$$ ##### 8.1.6.19. Answer. $$(3a+4b)(9a^2-12ab+16b^2)$$ ##### 8.1.6.21. Answer. $$(4t^3+w^2)(16t^6-4t^3w^2+w^4)$$ #### Applications ##### 8.1.6.23. Answer. 1. length: $$w+3\text{;}$$ height: $$w-2$$ 2. $$\displaystyle w^3+w^2-6w$$ 3. $$\displaystyle 5w^2+w-12$$ ##### 8.1.6.25. Answer. 1. $$\displaystyle \dfrac{2}{3}\pi r^3+\pi r^2h$$ 2. $$\displaystyle V(r)=\dfrac{14}{3}\pi r^3$$ ##### 8.1.6.27. Answer. 1. $$\displaystyle 0, 9$$ 2. $$0\le x \le 9\text{;}$$ $$R\ge 0$$ for these values 3. $$\dfrac{28}{3}$$ points 4. $$36$$ points 5. $$3$$ ml or $$8.2$$ ml ##### 8.1.6.29. Answer. 1. $$20$$ cm 2. $$100$$ cm ##### 8.1.6.31. Answer. 1. $$\displaystyle 6 + x + 5x^2$$ 2. $$\displaystyle 4 - 7x^2 - 8x^4$$ ### 8.2Algebraic Fractions8.2.5Problem Set 8.2 #### Warm Up ##### 8.2.5.1. Answer. 1. $$\dfrac{-3}{5} \text{,}$$ $$\dfrac{3}{7}$$ 2. $$\displaystyle 3$$ 3. $$\displaystyle -1$$ ##### 8.2.5.3. Answer. 1. $$\dfrac{-4}{3} \text{,}$$ $$\dfrac{40}{399}$$ 2. $$\displaystyle 1,~-1$$ 3. $$\displaystyle 0$$ ##### 8.2.5.5. Answer. 1. $$\displaystyle \dfrac{5}{x}$$ 2. $$\displaystyle \dfrac{12b}{7}$$ ##### 8.2.5.7. Answer. 1. $$\displaystyle \dfrac{-8}{5}$$ 2. $$\displaystyle \dfrac{a}{9}$$ #### Skills Practice ##### 8.2.5.9. Answer. None are correct ##### 8.2.5.11. Answer. 1. cannot be reduced 2. $$\displaystyle 1$$ 3. cannot be reduced 4. cannot be reduced ##### 8.2.5.13. Answer. (b) ##### 8.2.5.15. Answer. (a) ##### 8.2.5.17. Answer. $$\dfrac{a}{a-3}$$ ##### 8.2.5.19. Answer. $$\dfrac{1}{a+b}$$ ##### 8.2.5.21. Answer. $$\dfrac{y+3x}{y-3x}$$ ##### 8.2.5.23. Answer. $$y-2$$ ##### 8.2.5.25. Answer. $$\dfrac{-a}{a+1}$$ ##### 8.2.5.27. Answer. $$\dfrac{4z^2+6z+9}{2z+3}$$ #### Applications ##### 8.2.5.29. Answer. 1. 30 min 2. 50 min 3. 50 min 4. 6 mph 5. The time increases. If the current is 10 mph, the team will not be able to row upstream ##### 8.2.5.31. Answer. 1. $$\displaystyle 0\le p \lt 100$$ 2.  $$p$$ $$0$$ $$25$$ $$50$$ $$75$$ $$90$$ $$100$$ $$C$$ $$0$$ $$120$$ $$360$$ $$1080$$ $$3240$$ $$--$$ 3. 60% 4. $$\displaystyle p\lt 80\%$$ 5. $$p=100\text{:}$$ The cost of extracting more ore grows without bound as the amount extracted approaches 100%. ##### 8.2.5.33. Answer. 1. $$\dfrac{200}{2x-1}$$ square centimeters 2. 8; If $$x=13\text{,}$$ the area of the cross-section is 8 $$\text{cm}^2\text{.}$$ ##### 8.2.5.35. Answer. 1. $$4500+\dfrac{3000}{x} \text{;}$$ $$C(x) = 6x + 4500 + \dfrac{3000}{x}$$ 2.  $$x$$ $$20$$ $$40$$ $$60$$ $$80$$ $$100$$ $$C$$ $$4770$$ $$4815$$ $$5018$$ $$5130$$ 3.$4768.33
4. 22; 14>
5. The graph of $$C$$ approaches the line as an asymptote.

### 8.3Operations on Algebraic Fractions8.3.8Problem Set 8.3

#### Warm Up

##### 8.3.8.1.
1. $$\displaystyle \dfrac{4}{15}$$
2. $$\displaystyle \dfrac{2z}{3w}$$
##### 8.3.8.3.
1. $$\displaystyle 28$$
2. $$\displaystyle 28$$
##### 8.3.8.5.
1. $$\displaystyle \dfrac{1}{6}$$
2. $$\displaystyle \dfrac{1}{6y^2}$$

#### Skills Practice

##### 8.3.8.7.
1. $$\displaystyle \dfrac{-36a^2}{7}$$
2. $$\displaystyle \dfrac{8b}{3b+3}$$
3. $$\displaystyle \dfrac{v^2}{1-v^2}$$
##### 8.3.8.9.
$$\dfrac{1}{8x}$$
##### 8.3.8.11.
$$\dfrac{a(2a-1)}{a+4}$$
##### 8.3.8.13.
$$\dfrac{6x(x-2)(x-1)^2}{(x^2-8)(x^2-2x+4)}$$
##### 8.3.8.15.
$$\dfrac{9a^3}{14b^3}$$
##### 8.3.8.17.
$$\dfrac{3a}{2}$$
##### 8.3.8.19.
$$\dfrac{x^2}{y-1}$$
##### 8.3.8.21.
$$\dfrac{(z+2)^2}{z^2(2z-1)}$$
##### 8.3.8.23.
$$(x-y)(4x^2+2xy+y^2)$$
##### 8.3.8.25.
$$\dfrac{2x+y}{3x}$$
##### 8.3.8.27.
$$z-3$$
##### 8.3.8.29.
$$\dfrac{10x+3}{4x^2}$$
##### 8.3.8.31.
$$\dfrac{13}{8a-4b}$$
##### 8.3.8.33.
$$\dfrac{h^2+2h-3}{h+2}$$
##### 8.3.8.35.
$$\dfrac{6x-x^2-4}{x(x-2)}$$
##### 8.3.8.37.
$$\dfrac{19}{6(p-2)}$$
##### 8.3.8.39.
$$\dfrac{5k+1}{k(k-3)(k+1)}$$
##### 8.3.8.41.
$$\dfrac{3y-3y^2}{(y+1)(2y-1)}$$
##### 8.3.8.43.
$$12xy^2(x+y)^2$$
##### 8.3.8.45.
$$x(x-1)^3$$

#### Applications

##### 8.3.8.47.
$$\dfrac{4LR}{D^2}$$
##### 8.3.8.49.
$$\dfrac{2L}{c}+\dfrac{2LV^2}{c^3}$$
##### 8.3.8.51.
$$\dfrac{3q}{8\pi R}-\dfrac{a^2q}{8 \pi R^3}$$
##### 8.3.8.53.
$$-8t +\dfrac{1}{4} - \dfrac{3}{t}$$
##### 8.3.8.55.
1. $$\displaystyle \dfrac{x}{2}$$
2. $$\displaystyle 2x$$
3. $$\displaystyle \dfrac{1}{2x}$$
##### 8.3.8.57.
1. $$\displaystyle \dfrac{1}{a+b}$$
2. $$\displaystyle \dfrac{3}{4(a+b)}$$
3. $$\displaystyle \dfrac{4}{3(a+b)}$$
##### 8.3.8.59.
$$\dfrac{-H+ST}{RT}$$
##### 8.3.8.61.
$$\dfrac{4L-R^2C}{4L^2C}$$
##### 8.3.8.63.
$$\dfrac{2r^2_2ra+a^2}{a^2}$$
##### 8.3.8.65.
1. $$\dfrac{144}{x^2-2x}$$ sq ft
2. $$\displaystyle \dfrac{48x-48}{x^2-2x} ft$$
##### 8.3.8.67.
1. $$\dfrac{900}{400+w}$$ hr
2. $$\dfrac{900}{400-w}$$ hr
3. Orville, by $$\dfrac{1800w}{160,000-w^2}$$ hr

### 8.4More Operations on Fractions8.4.5Problem Set 8.4

#### Warm Up

##### 8.4.5.1.
$$\dfrac{2x^2+x-2}{x(x-1)}$$
##### 8.4.5.3.
$$\dfrac{x+2}{x-1}$$

#### Skills Practice

##### 8.4.5.5.
$$6y$$
##### 8.4.5.7.
$$\dfrac{5}{16}$$
##### 8.4.5.9.
$$\dfrac{7}{10a+2}$$
##### 8.4.5.11.
$$\dfrac{2x+1}{x}$$
##### 8.4.5.13.
$$\dfrac{nq}{p+q}$$
##### 8.4.5.15.
$$\dfrac{u-v}{xv}$$
##### 8.4.5.17.
$$\dfrac{1}{2}x^2\dfrac{1}{2}-\dfrac{1}{3x^2}$$
##### 8.4.5.19.
$$x-2+\dfrac{3}{y}$$
##### 8.4.5.21.
$$2y+5+\dfrac{2}{2y+1)}$$
##### 8.4.5.23.
$$4z^3-2z^2+3z+1+\dfrac{2}{2z+1}$$

#### Applications

##### 8.4.5.25.
$$\overline{PQ}$$ and $$overline{RS}: ~\dfrac{b}{a}\text{;}$$ $$\overline{QR}$$ and $$overline{SP}: ~\dfrac{b}{a}$$
##### 8.4.5.27.
1. $$\displaystyle \dfrac{1}{f}=\dfrac{2q+60}{q^2+60q}$$
2. $$\displaystyle f=\dfrac{q^2+60q}{2q+60}$$
##### 8.4.5.29.
$$\dfrac{n^2-k^2}{n^2}$$
##### 8.4.5.31.
$$\dfrac{2cd}{c^2-u^2}$$
##### 8.4.5.33.
$$\dfrac{KL}{N(L-F)}$$
##### 8.4.5.35.
$$\dfrac{1}{m+2h}$$
##### 8.4.5.37.
$$\dfrac{x^2+y^2}{x^2y^2}$$
##### 8.4.5.39.
$$\dfrac{b^2-a^2}{ab}$$
##### 8.4.5.41.
$$\dfrac{y}{y-x}$$
##### 8.4.5.43.
$$\dfrac{\sqrt{15}}{3}$$
##### 8.4.5.45.
$$\dfrac{2\sqrt{3}+\sqrt{6}}{9}$$
##### 8.4.5.47.
$$\dfrac{6\sqrt{3}+7\sqrt{2}}{5}$$

### 8.5Equations with Fractions8.5.7Problem Set 8.5

#### Warm Up

##### 8.5.7.1.
$$\dfrac{-1}{2}$$
##### 8.5.7.3.
$$\dfrac{13}{8}$$
##### 8.5.7.5.
$$\pm\sqrt{\dfrac{15}{8}}$$
##### 8.5.7.7.
$$\dfrac{1800}{1849}\approx 0.97$$

#### Skills Practice

##### 8.5.7.9.
$$-2,~1$$
##### 8.5.7.11.
$$-6$$
##### 8.5.7.13.
$$1$$
##### 8.5.7.15.
$$\dfrac{-14}{5}$$
##### 8.5.7.17.
We don’t multiply by the LCD in addition problems.
##### 8.5.7.19.
$$r=\dfrac{S-a}{a}$$
##### 8.5.7.21.
$$x=a-\dfrac{ay}{b}$$
##### 8.5.7.23.
$$R=\dfrac{Cr}{r-C}$$

#### Applications

2 mph
24 days
##### 8.5.7.29.
$$168=\dfrac{72p}{100-p}\text{;}$$ $$p=70\%$$
##### 8.5.7.31.
1. 0.268
2. $$\displaystyle \dfrac{44+x}{164+x}$$
3. 21
##### 8.5.7.33.
1. $$\displaystyle t=\dfrac{144}{v-20}$$
2. $$\displaystyle t=\dfrac{144}{v+20}$$
3. $$\displaystyle (100,3)$$
4. $$\displaystyle t=\dfrac{144}{v-20} + t=\dfrac{144}{v+20} = 3$$
5. 100 mph
##### 8.5.7.35.
$$28 \dfrac{1}{3}$$ miles
##### 8.5.7.37.
1. $$\displaystyle AE = 1,~ DE = x - 1,~ CD = 1$$
2. $$\displaystyle \dfrac{1}{x}=\dfrac{x-1}{x}$$
3. $$\displaystyle \dfrac{1+\sqrt{5}}{2}$$
##### 8.5.7.39.
Because $$x=1\text{,}$$ dividing by $$x-1$$ in the fourth step is dividing by $$0\text{.}$$
##### 8.5.7.41.
1. $$\displaystyle x=\dfrac{1}{2}$$

### 8.6Chapter 8 Summary and Review8.6.3Chapter 8 Review Problems

#### 8.6.3.1.

$$2x^3-11x^2+19x-10$$

#### 8.6.3.3.

$$(2x-3z)(4x^2+6xz+9z^2)$$

#### 8.6.3.5.

1. $$\displaystyle \dfrac{1}{6}n^3-\dfrac{1}{2}n^2+\dfrac{1}{3}n$$
2. $$\displaystyle 220$$
3. $$\displaystyle 20$$

#### 8.6.3.7.

1. $$\displaystyle V=\dfrac{\pi h^3}{4}$$
2. $$2\pi\text{ cm}^3 \approx 6.28\text{ cm}^3 \text{;}$$ $$16\pi\text{ cm}^3 \approx 50.27\text{ cm}^3$$

#### 8.6.3.9.

1. 338
2. Months 2 and 02
3. During month 6. The number of members eventually decreases to zero.

#### 8.6.3.11.

1. $$\displaystyle t_1=\dfrac{90}{v-2}$$
2. $$\displaystyle t_2=\dfrac{90}{v+2}$$
3. $$\displaystyle \dfrac{90}{v-2}+\dfrac{90}{v+2}=24$$
4. 8 mph

#### 8.6.3.13.

$$\dfrac{a}{2(a-1)}$$

#### 8.6.3.15.

$$\dfrac{y^2-2x}{2}$$

#### 8.6.3.17.

$$\dfrac{a-3}{2a+6)}$$

#### 8.6.3.19.

$$10ab$$

#### 8.6.3.21.

$$\dfrac{6x}{2x+3}$$

#### 8.6.3.23.

$$\dfrac{a^2-2a}{a^2+3a+2}$$

#### 8.6.3.25.

$$\dfrac{1}{2x-1}$$

#### 8.6.3.27.

$$9x^2-7+\dfrac{4}{x^2}-\dfrac{1}{x^4}$$

#### 8.6.3.29.

$$x^2-2x-2-\dfrac{1}{x-2}$$

#### 8.6.3.31.

$$\dfrac{2}{x}$$

#### 8.6.3.33.

$$\dfrac{3x+1}{2(x-3)(x+3)}$$

#### 8.6.3.35.

$$\dfrac{2a^2-a+1}{(a-3)(a-1{})}$$

#### 8.6.3.37.

$$\dfrac{1}{5}$$

#### 8.6.3.39.

$$\dfrac{x}{x+4}$$

#### 8.6.3.41.

$$-2$$

No solution

#### 8.6.3.45.

$$n=\dfrac{Ct}{C-V}$$

#### 8.6.3.47.

$$q=\dfrac{pr}{r-p}$$

#### 8.6.3.49.

$$\dfrac{x^3+y}{x^3y}$$

#### 8.6.3.51.

$$\dfrac{1-x^2y^2}{xy}$$

#### 8.6.3.53.

$$\dfrac{-(x-y)^2}{xy}$$

### 9Equations and Graphs9.1Properties of Lines9.1.4Problem Set 9.1

#### Warm Up

##### 9.1.4.1.
1. $$A:$$negative; $$B:$$ negative; $$C:$$ positive; $$D:$$ zero
2. $$B\text{,}$$ $$A\text{,}$$ $$D\text{,}$$ $$C$$
##### 9.1.4.3.
 Number Negativereciprocal Theirproduct $$\dfrac{2}{3}$$ $$\dfrac{-3}{2}$$ $$-1$$ $$\dfrac{-5}{2}$$ $$\dfrac{2}{5}$$ $$-1$$ $$6$$ $$\dfrac{-1}{6}$$ $$-1$$ $$-4$$ $$\dfrac{1}{4}$$ $$-1$$ $$-1$$ $$1$$ $$-1$$

#### Skills Practice

##### 9.1.4.5.
1. $$m$$ is undefined; $$(4,0)$$
##### 9.1.4.7.
$$x=-5$$
##### 9.1.4.9.
$$y=6$$
##### 9.1.4.11.
parallel: a, g, h; perpendicular: c, f
##### 9.1.4.13.
1. No
2. 3 and 3.1; no
3. 68. The two lines meet at $$(20,68) \text{.}$$

#### Applications

##### 9.1.4.15.
b. $$m_{PQ}=\dfrac{-5}{2} \text{;}$$ $$m_{PR}=\dfrac{2}{5}$$
##### 9.1.4.17.
1. $$y=\dfrac{3}{2}x+\dfrac{5}{2}\text{;}$$ the graph is below.
2. $$\displaystyle \dfrac{3}{2}$$
3. $$\displaystyle y=\dfrac{3}{2}x+\dfrac{7}{2}$$
##### 9.1.4.19.
1. $$\displaystyle y=-2x-8$$
2. $$\displaystyle y=\dfrac{1}{2}x-3$$
##### 9.1.4.21.
$$y=\dfrac{3}{2}x+\dfrac{15}{2}$$

### 9.2The Distance and Midpoint Formulas9.2.7Problem Set 9.2

#### Warm Up

1. False
2. False
##### 9.2.7.3.
 $$x$$ $$-4$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$y$$ $$0$$ $$\pm\sqrt{7}$$ $$\pm\sqrt{12}$$ $$\pm\sqrt{15}$$ $$\pm 4$$ $$\pm\sqrt{15}$$ $$\pm\sqrt{12}$$ $$\pm\sqrt{7}$$ $$0$$
##### 9.2.7.5.
$$\dfrac{5\pm \sqrt{33}}{2}$$

#### Skills Practice

##### 9.2.7.7.
distance: $$\sqrt{20} \text{;}$$ midpoint: $$\left(0,-2 \right)$$
##### 9.2.7.9.
distance: $$8\text{;}$$ midpoint: $$\left(-2,-1 \right)$$
##### 9.2.7.11.
center: $$(0,0) \text{;}$$ radius: $$5$$
##### 9.2.7.13.
center: $$(-3,0) \text{;}$$ radius: $$\sqrt{10}$$

#### Applications

##### 9.2.7.15.
$$15+\sqrt{80}+\sqrt{41} \approx 30.3$$
##### 9.2.7.17.
$$\sqrt{50} \approx 7.1$$
##### 9.2.7.19.
$$y=\dfrac{1}{2}x + \dfrac{5}{4}$$
##### 9.2.7.21.
1. $$\displaystyle (220,-38.5)$$
2. Both distances are 165.8 mi.
##### 9.2.7.29.
$$(x+4)^2+y^2=20 \text{;}$$ center: $$(-4,0) \text{;}$$ radius: $$\sqrt{20}$$
##### 9.2.7.31.
$$x^2+(y-5)^2=27 \text{;}$$ center: $$(0,5) \text{;}$$ radius: $$\sqrt{27}$$
##### 9.2.7.33.
$$\left(x-\dfrac{3}{2}\right)^2+(y+4)^2=\dfrac{29}{4}$$
##### 9.2.7.35.
$$\left(x+3\right)^2+(y+1)^2=1$$

### 9.3Conic Sections: Ellipses9.3.5Problem Set 9.3

#### Warm Up

##### 9.3.5.1.
1. $$\displaystyle x^2+y^2=r^2$$
2. $$\displaystyle \dfrac{x^2}{r^2}+\dfrac{y^2}{r^2}=1$$
3. $$\displaystyle a=b=r$$
##### 9.3.5.3.
$$(-2,6)$$
##### 9.3.5.5.
$$x$$-intercepts: $$(-6,0)\text{,}$$ $$(4,0)\text{,}$$ vertex: $$\left(-1,\dfrac{25}{2}\right)$$

#### Skills Practice

##### 9.3.5.13.
1. $$\displaystyle \dfrac{x^2}{9}+\dfrac{y^2}{4}=1$$
2.  $$x$$ $$0$$ $$\pm 3$$ $$-2$$ $$\dfrac{\pm 3\sqrt{3} }{2}$$ $$y$$ $$\pm 2$$ $$0$$ $$\dfrac{\pm 2\sqrt{5} }{2}$$ $$1$$
##### 9.3.5.15.
2. $$\displaystyle (-1,\pm \sqrt{3})$$
##### 9.3.5.17.
1. $$\displaystyle a=\sqrt{3},~b=\sqrt{6}$$
2. None
##### 9.3.5.19.
1. $$(-1,4)\text{,}$$ $$(7,4)\text{,}$$ $$(3,1)\text{,}$$ $$(3,7)$$ (Others are possible.)
##### 9.3.5.21.
1. $$\displaystyle \dfrac{x^2}{4}+\dfrac{(y-2)^2}{9}=1$$
##### 9.3.5.23.
1. $$\displaystyle \dfrac{(x-3)^2}{1}+\dfrac{(y+2)^2}{8}=1$$
##### 9.3.5.25.
$$\dfrac{(x-1)^2}{9}+\dfrac{(y-6)^2}{4}=1$$
##### 9.3.5.27.
$$\dfrac{(x-3)^2}{25}+\dfrac{(y-3)^2}{16}=1$$

#### Applications

##### 9.3.5.29.
1. $$\displaystyle \dfrac{x^2}{10^2}+\dfrac{y^2}{7^2} = 1$$
2. 4.2 ft
##### 9.3.5.31.
1. $$\displaystyle \dfrac{x^2}{24^2}+\dfrac{y^2}{8^2} = 1$$
2. 10.29 ft

### 9.4Conic Sections: Hyperbolas9.4.6Problem Set 9.4

#### Skills Practice

##### 9.4.6.9.
1. $$\displaystyle \dfrac{x^2}{9} - \dfrac{y^2}{3} =1$$
2.  $$x$$ $$0$$ $$\dfrac{\pm3\sqrt{5}}{2}$$ $$5$$ $$\dfrac{\pm 15}{4}$$ $$y$$ undefined $$\pm 2$$ $$\dfrac{\pm 16}{3}$$ $$-3$$
##### 9.4.6.13.
1. $$\left(-2, -2\pm \sqrt{6} \right) \text{,}$$ $$\left(-2 \pm \sqrt{5} ,1\right)$$
##### 9.4.6.15.
1. $$\left(1 -4\pm 2\sqrt{2} \right) \text{,}$$ $$(-2,-8) \text{,}$$ $$(4,-8)$$
##### 9.4.6.17.
1. $$\left(3 \pm \sqrt{5} \right) \text{,}$$ $$\left(3 \pm 2\sqrt{2}, 3 \right)$$
##### 9.4.6.19.
$$16x^2-y^2-192x-4y+556=0$$
##### 9.4.6.21.
$$x^2-4y^2+10x-16y-27=0$$
##### 9.4.6.23.
Parabola; vertex $$(0,2)\text{,}$$ opens downward, $$a=\dfrac{-1}{2}$$
##### 9.4.6.25.
Hyperbola; center $$\left(\dfrac{-1}{8},-1 \right) \text{,}$$ transverse axis vertical, $$a^2=\dfrac{15}{65} \text{,}$$ $$b^2=\dfrac{15}{16}$$
##### 9.4.6.27.
Parabola; vertex $$(-4,2)\text{,}$$ opens upward, $$a=\dfrac{1}{4}$$

520 ft
472.5 ft

### 9.5Nonlinear Systems9.5.4Problem Set 9.5

#### Warm Up

##### 9.5.4.1.
$$(1,-2)$$
##### 9.5.4.3.
$$x=3, 16$$

#### Skills Practice

##### 9.5.4.5.
$$(-1, 12), (4, 7)$$
No solution
##### 9.5.4.9.
$$(1, 4)$$
##### 9.5.4.11.
$$(2,2) \text{,}$$ $$(-2,-2)$$
##### 9.5.4.13.
$$(2,-1) \text{,}$$ $$(-2,1) \text{,}$$ $$(1,-2) \text{,}$$ $$(-1,2)$$
##### 9.5.4.15.
$$(\pm 2,-\pm 5)$$
##### 9.5.4.17.
$$(\pm 6,-\pm 2)$$
##### 9.5.4.19.
$$(0,4) \text{,}$$ $$(-2,0)$$

#### Applications

12 ft by 18 ft
##### 9.5.4.23.
$$P=6$$ lb per sq in; $$V=5$$ cu. in.
##### 9.5.4.25.
1. 1000 or 5000
2. 2000 or 4000
3. harvst 1800; stable population 3000
4. extinction
##### 9.5.4.27.
1. $$\displaystyle (200,~ 2600), (1400,~ 18,200)$$
2. $$\displaystyle x=800$$

### 9.6Chapter 9 Summary and Review9.6.3Chapter 9 Review Problems

parallel

perpendicular

#### 9.6.3.3.

$$y=\dfrac{-2}{3}x+\dfrac{14}{3}$$

#### 9.6.3.4.

$$y=\dfrac{3}{2}x+\dfrac{5}{2}$$

#### 9.6.3.5.

$$y=\dfrac{2}{3}x-\dfrac{26}{3}$$

#### 9.6.3.6.

$$y=\dfrac{3}{2}x-2$$

21.59; yes

10.8

#### 9.6.3.19.

1. $$\displaystyle (x-2)^2+(y+1)^2=9$$
2. Circle: center $$(2,-1)\text{,}$$ radius 3

#### 9.6.3.20.

1. $$\displaystyle x^2+(y-3)^2=13$$
2. Circle: center $$(0,3)\text{,}$$ radius $$\sqrt{13}$$

#### 9.6.3.21.

1. $$\displaystyle \dfrac{(x-2)^2}{4}+\dfrac{(y+2)^2}{16}=1$$
2. Ellipse: center $$(2,-2)\text{,}$$ $$a=2\text{,}$$ $$b=4$$

#### 9.6.3.22.

1. $$\displaystyle \dfrac{(x+1)^2}{5}+\dfrac{(y-2)^2}{8}=1$$
2. Ellipse: center $$(-1,2)\text{,}$$ $$a=\sqrt{5}\text{,}$$ $$b=\sqrt{8}$$

#### 9.6.3.23.

1. $$\displaystyle y+10=(x-4)^2$$
2. Parabola: vertex $$(4,10)\text{,}$$ opens upward, $$a=1$$

#### 9.6.3.24.

1. $$\displaystyle x-2=\dfrac{-1}{4}(y+3)^2$$
2. Parabola: vertex $$(2,-3)\text{,}$$ opens left, $$a=\dfrac{-1}{4}$$

#### 9.6.3.25.

1. $$\displaystyle y+2=-(x-2)^2$$
2. Parabola: vertex $$(2,-2)\text{,}$$ opens downward, $$a=-1$$

#### 9.6.3.26.

1. $$\displaystyle x+\dfrac{3}{2}=\dfrac{1}{2}(y-1)^2$$
2. Parabola: vertex $$\left(\dfrac{-3}{2},1\right)\text{,}$$ opens right, $$a=\dfrac{1}{2}$$

#### 9.6.3.27.

1. $$\displaystyle \dfrac{(y-4)^2}{6}-\dfrac{(x+2)^2}{4}=1$$
2. Hyperbola: center $$(-2,4)\text{,}$$ transverse axis vertical, $$a=2\text{,}$$ $$b=\sqrt{6}$$

#### 9.6.3.28.

1. $$\displaystyle \dfrac{(x-4)^2}{4}-\dfrac{(y+3)^2}{9}=1$$
2. Hyperbola: center $$(4,-3)\text{,}$$ transverse axis horizontal, $$a=2\text{,}$$ $$b=3$$

#### 9.6.3.29.

1. $$\displaystyle \dfrac{x^2}{5}-\dfrac{(y-3)^2}{10}=1$$
2. Hyperbola: center $$(0,3)\text{,}$$ transverse axis horizontal, $$a=\sqrt{5}\text{,}$$ $$b=\sqrt{10}$$

#### 9.6.3.30.

1. $$\displaystyle \dfrac{y^2}{3}-\dfrac{(x-4)^2}{12}=1$$
2. Hyperbola: center $$(4,0)\text{,}$$ transverse axis vertical, $$a=2\sqrt{3}\text{,}$$ $$b=\sqrt{310}$$

#### 9.6.3.31.

1. $$\displaystyle \dfrac{x^2}{25}+\dfrac{y^2}{64}=1$$
2. $$\displaystyle \dfrac{\pm 24}{5}$$

#### 9.6.3.32.

1. $$\displaystyle \dfrac{x^2}{169}+\dfrac{y^2}{81}=1$$
2. $$\displaystyle \dfrac{\pm 45}{13}$$

#### 9.6.3.33.

$$(x+4)^2+(y-3)^2=20$$

#### 9.6.3.34.

$$(x+2)^2+(y-4)^2=13$$

#### 9.6.3.35.

$$\dfrac{(x+1)^2}{16}+\dfrac{(y-4)^2}{4}=1$$

#### 9.6.3.36.

$$\dfrac{(x-3)^2}{4}+\dfrac{(y-1)^2}{25}=1$$

#### 9.6.3.37.

$$\dfrac{(x-2)^2}{16}-\dfrac{(y+3)^2}{9}=1$$

#### 9.6.3.38.

$$(x+3)^2 -\dfrac{(y-1)^2}{9}=1$$

#### 9.6.3.39.

$$(\pm 2, \pm 3)$$

#### 9.6.3.40.

$$(\pm\sqrt{3}, \pm 4)$$

#### 9.6.3.41.

$$(1,-2) \text{,}$$ $$(-1,2) \text{,}$$ $$\left(2\sqrt{3}, \dfrac{-1}{\sqrt{3}} \right) \text{,}$$ $$\left(-2\sqrt{3}, \dfrac{1}{\sqrt{3}} \right)$$

#### 9.6.3.42.

$$\left(\dfrac{\sqrt{34}}{2}, \dfrac{\sqrt{34}}{2} \right) \text{,}$$ $$\left(\dfrac{-\sqrt{34}}{2}, \dfrac{-\sqrt{34}}{2} \right)$$

#### 9.6.3.43.

Moia: 45 mph, Fran: 50 mph

12 in by 1 in

7 cm by 10 cm

7 ft by 2 ft

#### 9.6.3.47.

Morning train: 20 mph, evening train: 30 mph

#### 9.6.3.48.

Amount: $800, rate: 4% ### 10Logarithmic Functions10.1Logarithmic Functions10.1.6Problem Set 10.1 #### Warm Up ##### 10.1.6.1. Answer. $$9^y=729$$ ##### 10.1.6.3. Answer. $$10^{-4.5}=C$$ ##### 10.1.6.5. Answer. $$\log_b{\dfrac{1}{4}}$$ ##### 10.1.6.7. Answer. $$\log_{10}{\sqrt{\dfrac{xy}{z^3}}}$$ #### Skills Practice ##### 10.1.6.9. Answer. ##### 10.1.6.11. Answer. 1. $$\displaystyle 15.6144$$ 2. $$\displaystyle 0.4186$$ ##### 10.1.6.13. Answer. $$-1.58 \times 10^{-5}$$ ##### 10.1.6.15. Answer. 1. $$\displaystyle 25.70$$ 2. $$\displaystyle 3.31$$ ##### 10.1.6.17. Answer. 1. $$\displaystyle \dfrac{1}{2}$$ 2. $$\displaystyle 01$$ ##### 10.1.6.19. Answer. 1. 81 2. 4 3. 1.8 4. $$\displaystyle a$$ ##### 10.1.6.21. Answer. 1. IV 2. V 3. I 4. II 5. III 6. VI ##### 10.1.6.23. Answer. 1. 10,000 2. 100,000,000 ##### 10.1.6.25. Answer. $$4$$ ##### 10.1.6.27. Answer. 3 #### Applications ##### 10.1.6.29. Answer. 1. The graph resembles a logarithmic function. The function is close to the points but appears too steep at first and not steep enough after $$n = 15\text{.}$$ Overall, it is a good fit. 2. $$f$$ grows (more and more slowly) without bound. $$f$$ will eventually exceed $$100$$ per cent, but no one can forget more than 100% of what is learned. ##### 10.1.6.31. Answer. 1962 ### 10.2Logarithmic Scales10.2.6Problem Set 10.2 #### Warm Up ##### 10.2.6.1. Answer. 1. 0 and 1 2. 2 and 3 3. $$-1$$ and 0 4. 6 and 7 ##### 10.2.6.3. Answer. 1. 3981.1 2. 5.01 3. 0.00079 4. 0.398 #### Skills Practice ##### 10.2.6.5. Answer. ##### 10.2.6.7. Answer. ##### 10.2.6.9. Answer. $$1.58\text{,}$$ $$6.31\text{,}$$ $$15.8\text{,}$$ $$63.1$$ ##### 10.2.6.11. Answer. $$3.2$$ ##### 10.2.6.13. Answer. $$0.0126$$ ##### 10.2.6.15. Answer. $$100$$ ##### 10.2.6.17. Answer. 6,309,573 watts per square meter #### Applications ##### 10.2.6.19. Answer. $$1\text{,}$$ $$80\text{,}$$ $$330\text{,}$$ $$1600\text{,}$$ $$7000\text{,}$$ $$4\times 10^7$$ ##### 10.2.6.21. Answer. Proxima Centauri: $$15.5\text{;}$$ Barnard: $$13.2\text{;}$$ Sirius: $$1.4\text{;}$$ Vega: $$0.6\text{;}$$ Arcturus: $$-0.4\text{;}$$ Antares: $$-4.7\text{;}$$ Betelgeuse: $$-7.2$$ ##### 10.2.6.23. Answer. ##### 10.2.6.25. Answer. $$10^{3.4} \approx 2512$$ ##### 10.2.6.27. Answer. A: $$a\approx 45\text{,}$$ $$p \approx 7.4\%\text{;}$$ B: $$a \approx 400\text{,}$$ $$p \approx 15\%\text{;}$$ C: $$a\approx 6000\text{,}$$ $$p\approx 50\%\text{;}$$ D: $$a \approx 13000\text{,}$$ $$p \approx 45\%$$ ##### 10.2.6.29. Answer. 12.6 ##### 10.2.6.32. Answer. $$3160$$ ##### 10.2.6.33. Answer. $$\approx 25,000$$ ### 10.3The Natural Base10.3.6Homework 10.3 #### Skills Practice ##### 10.3.6.1. Answer.  $$x$$ $$-10$$ $$-5$$ $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ $$f(x)$$ $$0.135$$ $$0.368$$ $$1$$ $$2.718$$ $$7.389$$ $$20.086$$ $$54.598$$ ##### 10.3.6.3. Answer.  $$x$$ $$-10$$ $$-5$$ $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ $$f(x)$$ $$20.086$$ $$4.482$$ $$1$$ $$0.223$$ $$0.05$$ $$0.011$$ $$0.00248$$ ##### 10.3.6.5. Answer. 1. $$\displaystyle 2$$ 2. $$\displaystyle 5t$$ 3. $$\displaystyle \dfrac{1}{x}$$ 4. $$\displaystyle \dfrac{1}{2}$$ ##### 10.3.6.7. Answer. 1. $$\displaystyle 0.64$$ 2. $$\displaystyle 3.81$$ 3. $$\displaystyle -1.20$$ ##### 10.3.6.9. Answer. 1. $$\displaystyle 4.14$$ 2. $$\displaystyle 1.88$$ 3. $$\displaystyle 0.07$$ ##### 10.3.6.11. Answer. $$P (t) = 20\left(e^{0.4} \right)^t \approx 20\cdot 1.492^t\text{;}$$ increasing; initial value $$20$$ ##### 10.3.6.13. Answer. $$P (t) = 6500\left(e^{-2.5} \right)^t \approx 6500\cdot 0.082^t\text{;}$$ decreasing; initial value $$6500$$ ##### 10.3.6.15. Answer. 1.  $$x$$ $$0$$ $$0.5$$ $$1$$ $$1.5$$ $$2$$ $$2.5$$ $$e^x$$ $$1$$ $$1.6487$$ $$2.7183$$ $$4.4817$$ $$7.3891$$ $$12.1825$$ 2. Each ratio is $$e^{0.5} \approx 1.6487\text{:}$$ Increasing $$x$$-values by a constant $$\Delta x = 0.5$$ corresponds to multiplying the $$y$$-values of the exponential function by a constant factor of $$e^{\Delta x}\text{.}$$ ##### 10.3.6.17. Answer. 1.  $$x$$ $$0$$ $$0.6931$$ $$1.3863$$ $$2.0794$$ $$2.7726$$ $$3.4657$$ $$4.1589$$ $$e^x$$ $$1$$ $$2$$ $$4$$ $$8$$ $$16$$ $$32$$ $$64$$ 2. Each difference in $$x$$-values is approximately $$\ln 2\approx 0.6931\text{:}$$ Increasing $$x$$-values by a constant $$\Delta x = \ln 2$$ corresponds to multiplying the $$y$$-values of the exponential function by a constant factor of $$e^{\Delta x} = e^{\ln 2} = 2\text{.}$$ That is, each function value is approximately equal to double the previous one. ##### 10.3.6.19. Answer. $$0.8277$$ ##### 10.3.6.21. Answer. $$-2.9720$$ ##### 10.3.6.23. Answer. $$1.6451$$ ##### 10.3.6.25. Answer. $$-3.0713$$ ##### 10.3.6.27. Answer. $$t=\dfrac{1}{k}\ln y$$ ##### 10.3.6.29. Answer. $$t=\ln \left(\dfrac{k}{k-y}\right)$$ ##### 10.3.6.31. Answer. $$k=e^{T/T_0}-10$$ ##### 10.3.6.33. Answer. 1.  $$n$$ $$0.39$$ $$3.9$$ $$39$$ $$390$$ $$\ln n$$ $$-0.942$$ $$1.361$$ $$3.664$$ $$5.966$$ 2. Each difference in function values is approximately $$\ln 10\approx 2.303\text{:}$$ Multiplying $$x$$-values by a constant factor of 10 corresponds to adding a constant value of $$\ln 10$$ to the $$y$$-values of the natural log function. ##### 10.3.6.35. Answer. 1.  $$n$$ $$2$$ $$4$$ $$8$$ $$16$$ $$\ln n$$ $$0.693$$ $$1.386$$ $$2.079$$ $$2.773$$ 2. Each quotient equals $$k\text{,}$$ where $$n = 2^k\text{.}$$ Because $$\ln n = \ln 2^k = k\cdot \ln 2\text{,}$$ $$k = \dfrac{\ln n}{\ln 2}\text{.}$$ ##### 10.3.6.37. Answer. 1. $$\displaystyle N (t) = 100e^{(\ln 2)t}\approx 100e^{0.6931t}$$ ##### 10.3.6.39. Answer. 1. $$\displaystyle N (t) = 1200e^{(\ln 0.6)t}\approx 1200e^{-0.5108t}$$ ##### 10.3.6.41. Answer. 1. $$\displaystyle N (t) = 10e^{(\ln 1.15)t}\approx 10e^{0.1398t}$$ #### Applications ##### 10.3.6.43. Answer. 1. $$\displaystyle N(t)=6000e^{0.04t}$$ 2.  $$t$$ $$0$$ $$5$$ $$10$$ $$15$$ $$20$$ $$25$$ $$30$$ $$N(t)$$ $$6000$$ $$7328$$ $$8951$$ $$10,933$$ $$13,353$$ $$16,310$$ $$19,921$$ 3. 15,670 4. 70.3 hrs ##### 10.3.6.45. Answer. 1. 941.8 lumens 2. 2.2 cm ##### 10.3.6.47. Answer. 1. 20,000 2. $$\displaystyle \left(\dfrac{35,000}{20,000} \right)^{1/10}\approx e^{0.056}$$ 3. $$\displaystyle P(t) = 20,000e^{0.056t}$$ 4. 107,188 ##### 10.3.6.49. Answer. 1. $$\displaystyle \left(\dfrac{385}{500} \right)^{1/2}\approx e^{-0.1307}$$ 2. $$\displaystyle N(t) = 500e^{-0.1307t}$$ 3. 135.3 mg ##### 10.3.6.51. Answer. 1. $$\displaystyle A(t) = 500e^{0.095t}$$ 2. 7.3 years 3. 7.3 years d–e ##### 10.3.6.53. Answer. 1. 6 hours 2. 6 hours ##### 10.3.6.55. Answer. 1. $$\frac{1}{2}N_0 \text{,}$$ $$\frac{1}{4}N_0 \text{,}$$ $$\frac{1}{16}N_0$$ 2. $$\displaystyle N (t) = N_0e^{-0.0866t}$$ ##### 10.3.6.57. Answer. 1. $$\displaystyle y = 116 (0.975)^t$$ 2. $$\displaystyle G (t) = 116e^{-0.025t}$$ 3. 28 minutes ### 10.4Chapter 10 Summary and Review10.4.3Chapter 10 Review Problems #### 10.4.3.5. Answer. $$-1$$ #### 10.4.3.7. Answer. $$\dfrac{1}{2}$$ #### 10.4.3.9. Answer. $$4$$ #### 10.4.3.11. Answer. $$\dfrac{-15}{8}$$ #### 10.4.3.13. Answer. $$\dfrac{9}{4}$$ #### 10.4.3.15. Answer. $$3$$ #### 10.4.3.17. Answer. $$x\approx 1.548$$ #### 10.4.3.19. Answer. $$x\approx 411.58$$ #### 10.4.3.21. Answer. $$x\approx 2.286$$ #### 10.4.3.23. Answer. $$\sqrt{x}$$ #### 10.4.3.25. Answer. $$k-3$$ #### 10.4.3.27. Answer. 1. $$\displaystyle P = 7,894,862e^{-0.011t}$$ 2. 1.095% #### 10.4.3.29. Answer. 1.$1419.07
2. 13.9 years
3. $$\displaystyle t = 20 \ln\left(\dfrac{A}{1000} \right)$$

#### 10.4.3.31.

$$t=\dfrac{-1}{k}\ln\left(\dfrac{y-6}{12} \right)$$

#### 10.4.3.33.

$$M=N^{Qt}$$

#### 10.4.3.35.

$$P (t) = 750 (1.3771)^t$$

#### 10.4.3.37.

$$N(t) = 600 e^{-0.9163t}$$