The slope-intercept form is useful when we know the initial value and the rate of change.

The point-slope form is useful when we know the rate of change and one point on the line.

.

Exercises1.6.3Chapter 1 Review Problems

1.

Last year, Pinwheel Industries introduced a new model calculator. It cost $2000 to develop the calculator and $20 to manufacture each one.

Complete the table of values showing the total cost, \(C\text{,}\) of producing \(n\) calculators.

\(n\)

\(100\)

\(500\)

\(800\)

\(1200\)

\(1500\)

\(C\)

\(\hphantom{0000}\)

\(\hphantom{0000}\)

\(\hphantom{0000}\)

\(\hphantom{0000}\)

\(\hphantom{0000}\)

Write an equation that expresses \(C\) in terms of \(n\text{.}\)

Graph the equation by hand.

What is the cost of producing 1000 calculators? Illustrate this as a point on your graph.

How many calculators can be produced for $10,000? Illustrate this as a point on your graph.

Answer.

\(n\)

\(100\)

\(500\)

\(800\)

\(1200\)

\(1500\)

\(C\)

\(4000\)

\(12,000\)

\(18,000\)

\(26,000\)

\(32,000\)

\(\displaystyle C = 20n + 2000\)

$22,000

400

2.

The world’s oil reserves were 2100 billion barrels in 2005; total annual consumption is 28 billion barrels per year.

Complete the table of values that shows the remaining oil reserves \(R\) in terms of time \(t\) (in years since 2005).

\(t\)

\(~ 5 ~\)

\(~10~\)

\(~15~\)

\(~20~\)

\(~25~\)

\(R\)

\(\)

\(\)

\(\)

\(\)

\(\)

Write an equation that expresses \(R\) in terms of \(t\text{.}\)

Find the intercepts and graph the equation by hand.

What do the intercepts tell us about the world’s oil supply?

Answer.

\(t\)

\(~ 5 ~\)

\(~10~\)

\(~15~\)

\(~20~\)

\(~25~\)

\(R\)

\(1960\)

\(1820\)

\(1680\)

\(1540\)

\(1400\)

\(\displaystyle R = 2100 - 28t\)

\((75, 0)\text{,}\)\((0, 2100)\)

\(t\)-intercept: The oil reserves will be gone in 2080; \(R\)-intercept: There were \(2100\) billion barrels of oil reserves in 2005.

3.

Alida plans to spend part of her vacation in Atlantic City and part in Saint-Tropez. She estimates that after airfare her vacation will cost $60 per day in Atlantic City and $100 per day in Saint-Tropez. She has $1200 to spend after airfare.

Write an equation that relates the number of days, \(C\text{,}\) Alida can spend in Atlantic City and the number of days, \(T\text{,}\) in Saint-Tropez.

Find the intercepts and graph the equation by hand.

If Alida spends 10 days in Atlantic City, how long can she spend in Saint-Tropez?

What do the intercepts tell us about Alida’s vacation?

Answer.

\(\displaystyle 60C+100T=1200\)

\((20,0) \text{,}\)\((0,12) \)

6 days

She can spend 20 days in Atlantic City if she spends no time in Saint-Tropez, or 12 days in Saint-Tropez if she spends no time in Atlantic City.

Exercise Group.

For Problems 4–9, graph the equation on graph paper. Use the most convenient method for each problem.

4.

\(\dfrac{x}{6}-\dfrac{y}{12}=1 \)

Answer.

5.

\(50x=40y-20,000 \)

Answer.

6.

\(1.4x+2.1y=8.4 \)

Answer.

7.

\(3x-4y=0 \)

Answer.

8.

\(3x-4y=0 \)

Answer.

9.

\(4x=-12 \)

Answer.

10.

The table shows the amount of oil, \(B\) (in thousands of barrels), left in a tanker \(t\) minutes after it hits an iceberg and springs a leak.

\(t\)

\(0\)

\(10\)

\(20\)

\(30\)

\(B\)

\(800\)

\(750\)

\(700\)

\(650\)

Write a linear function for \(B\) in terms of \(t\text{.}\)

Choose appropriate window settings on your calculator and graph your function.

Give the slope of the graph, including units, and explain the meaning of the slope in terms of the oil leak.

Answer.

\(\displaystyle B = 800 - 5t\)

\(m = -5\) thousand barrels/minute: The amount of oil in the tanker is decreasing by 5000 barrels per minute.

11.

An interior decorator bases her fee on the cost of a remodeling job. The accompanying table shows her fee, \(F\text{,}\) for jobs of various costs, \(C\text{,}\) both given in dollars.

\(C\)

\(5000\)

\(10,000\)

\(20,000\)

\(50,000\)

\(F\)

\(1000\)

\(1500\)

\(2500\)

\(5500\)

Write a linear function for \(F\) in terms of \(C\text{.}\)

Choose appropriate window settings on your calculator and graph your function.

Give the slope of the graph, including units, and explain the meaning of the slope in terms of the the decorator’s fee.

Answer.

\(\displaystyle F = 500 + 0.10C\)

\(m = 0.10\text{:}\) The fee increases by $0.10 for each dollar increase in the remodeling job.

Exercise Group.

For Problems 12–15, find the slope of the line segment joining the points.

12.

\((-1,4), ~(3,-2) \)

Answer.

\(\dfrac{-3}{2} \)

13.

\((5,0), ~(2,-6) \)

Answer.

\(2 \)

14.

\((6.2,1.4), ~(-2.3,4,8) \)

Answer.

\(-0.4 \)

15.

\((0,-6.4), ~(-5.6,3.4) \)

Answer.

\(-1.75\)

Exercise Group.

For Problems 16–17, which tables could describe variables related by a linear equation?

16.

\(r\)

\(E\)

\(1\)

\(5\)

\(2\)

\(\dfrac{5}{2} \)

\(3\)

\(\dfrac{5}{3} \)

\(4\)

\(\dfrac{5}{4} \)

\(5\)

\(1\)

\(s\)

\(t\)

\(10\)

\(6.2\)

\(20\)

\(9.7 \)

\(30\)

\(12.6 \)

\(40\)

\(15.8 \)

\(50\)

\(19.0\)

Answer.

neither

17.

\(w\)

\(A\)

\(2\)

\(-13\)

\(4\)

\(-23 \)

\(6\)

\(-33 \)

\(8\)

\(-43 \)

\(10\)

\(-53\)

\(x\)

\(G\)

\(0\)

\(0\)

\(2\)

\(5 \)

\(4\)

\(10 \)

\(8\)

\(20 \)

\(16\)

\(40\)

Answer.

both

Exercise Group.

For Problems 18–19, the table gives values for a linear equation in two variables. Fill in the missing values.

18.

\(d\)

\(V\)

\(-5\)

\(-4.8\)

\(-2\)

\(-3 \)

\(\)

\(-1.2\)

\(6\)

\(1.8 \)

\(10\)

\(\)

Answer.

\(d\)

\(V\)

\(-5\)

\(-4.8\)

\(-2\)

\(-3 \)

\(1\)

\(-1.2\)

\(6\)

\(1.8 \)

\(10\)

\(4.2\)

19.

\(q\)

\(S\)

\(-8\)

\(-8\)

\(-4\)

\(36 \)

\(3\)

\(\)

\(\)

\(200 \)

\(9\)

\(264\)

Answer.

\(q\)

\(S\)

\(-8\)

\(-8\)

\(-4\)

\(36 \)

\(3\)

\(168\)

\(5\)

\(200 \)

\(9\)

\(264\)

20.

The planners at AquaWorld want the small water slide to have a slope of 25%. If the slide is 20 feet tall, how far should the end of the slide be from the base of the ladder?

Answer.

80 ft

Exercise Group.

For Problems 21–24, find the slope and \(y\)-intercept of the line.

21.

\(2x-4y=5\)

Answer.

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

22.

\(\dfrac{1}{2} x+\dfrac{2}{3} y=\dfrac{5}{6} \)

Answer.

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

23.

\(8.4x + 2.1y = 6.3\)

Answer.

\(m=-4 \text{,}\)\(b= 3 \)

24.

\(y - 3 = 0 \)

Answer.

\(m=0 \text{,}\)\(b= 3 \)

Exercise Group.

For Problems 25 and 26,

Graph by hand the line that passes through the given point with the given slope.

The rate at which air temperature decreases with altitude is called the lapse rate. In the troposphere, the layer of atmosphere that extends from the Earth’s surface to a height of about 7 miles, the lapse rate is about \(3.6\degree\text{F}\) for every 1000 feet. (Source: Ahrens, 1998)

If the temperature on the ground is \(62\degree\text{F}\text{,}\) write an equation for the temperature, \(T\text{,}\) at an altitude of \(h\) feet.

What is the temperature outside an aircraft flying at an altitude of 30,000 feet? How much colder is that than the ground temperature?

What is the temperature at the top of the troposphere?