Because polar ice reflects light from the sun, the radiation balance over an ice-covered ocean is very different from the balance over an open ocean. The ice component of the climate system, called the cryosphere, plays an important role in the Earthβs radiation balance.
Climate models predict that global warming over the next few decades will occur mainly in the polar regions. As polar ice begins to melt, less sunlight is reflected into space, which raises the overall temperature and fuels further melting. This process is called ice albedo feedback. Since satellite monitoring began in 1979, Arctic sea ice cover has decreased about 10% per decade, falling to a startling new low in 2007.
Numerous factors influence the freezing point of sea water, including its salinity, or mineral content. In this Lesson weβll develop a formula for the freezing temperature of water in terms of its salinity.
Recall Practice 4 from Section 1.4: The graph shows the amount of garbage \(G\text{,}\) in tons, that has been deposited at a dumpsite \(t\) years after new regulations go into effect.
State the vertical intercept and the slope of the graph.
The slope tells us that garbage is accumulating at a rate of 12.5 tons per year. The vertical intercept tells us that when the new regulations went into effect, the dump held 25 tons of garbage.
Delbert decides to use DSL for his Internet service. Earthlink charges a $99 activation fee and $39.95 per month, DigitalRain charges $50 for activation and $34.95 per month, and FreeAmerica charges $149 for activation and $34.95 per month.
Using the formula \(m = \dfrac{\Delta y}{\Delta x}\text{,}\) it is fairly easy to calculate the slope of a line from its graph if there are two obvious points to use. However, if the coordinates are not easy, or if we donβt have a graph, we may need another method.
For example, if you walk from \(3^{\text{rd}}\) street to \(8^{\text{th}}\) street, your distance, \(s\text{,}\) from the center of town has increased by 5 blocks, or
\begin{gather*}
\Delta s = 8 - 3 = 5
\end{gather*}
If the temperature \(T\) drops from \(28\degree\) to \(22\degree\text{,}\) it has decreased by \(6\degree\text{,}\) or
\begin{gather*}
\Delta T = 22-28 = -6
\end{gather*}
The net change is positive if the variable increases, and negative if it decreases. For the graph shown at right, the net change in \(t\)-coordinate from \(P\) to \(Q\) is
\begin{equation*}
\Delta t = 51 - (-23) = 74
\end{equation*}
We can use the notion of net change to write a coordinate formula for computing slope. We will need subscripts to designate the coordinates of two different points. Weβll write \((x_1, y_1)\) for the coordinates of the first point, and \((x_2, y_2)\) for the coordinates of the second point.
Do not confuse subscripts with exponents! An exponent changes the value of a variable, so that for instance if \(x=3\) then \(x^2 = 3^2 = 9\text{,}\) but a subscript merely tells us which point the variable comes from, so that \(x_2\) just means "the \(x\)-coordinate of the second point."
Notice that the numerator of the slope formula, \(y_2 - y_1\text{,}\) gives the net change in \(y\text{,}\) or \(\Delta y\text{,}\) and the denominator, \(x_2 - x_1\text{,}\) gives the net change in \(x\text{,}\) or \(\Delta x\text{.}\) The coordinate formula is equivalent to our definition of slope, \(m=\dfrac{\Delta y}{\Delta x}\text{.}\)
The graph shows wine consumption, \(W\text{,}\) in the US, in millions of cases, starting in 1990. In 1993, Americans drank 188.6 million cases of wine. In 2003, Americans drank 258.3 million cases of wine.
Find the slope of the graph from 1993 to 2003. \(~\alert{\text{[TK]}}\)
If \(t = 0\) in 1990, then in 1993, \(t = 3\text{,}\) and in 2003, \(t = 13 \text{.}\) Thus, the points \(P(3, 188.6)\) and \(Q(13, 258.3)\) lie on the line. We want to compute the slope,
between these two points. Think of moving from \(P\) to \(Q\) in two steps, first moving horizontally to the right from \(P\) to the point \(R\text{,}\) and then vertically from \(R\) to \(Q\text{.}\) The coordinates of \(R\) are \((13, 188.6)\text{.}\) (Do you see why?) Then
\begin{align*}
\Delta t \amp= t_2-t_1 = 13-3=10\\
\Delta W \amp= W_2-W_1= 258.3-188.6=69.7
\end{align*}
In 1991, there were 64.6 burglaries per 1000 households in the United States. The number of burglaries reported annually declined at a roughly constant rate over the next decade, and in 2001 there were 28.7 burglaries per 1000 households. (Source: U.S. Department of Justice)
Give the coordinates of two points on the graph of \(B=mt+b\text{,}\) where \(t = 0\) in 1990, and \(B\) stands for the number of burglaries per 1000 households.
Now weβll consider using the slope formula for a different problem. If we know the slope of a line and the coordinates of one point on the line, we can use the coordinate formula for slope to find the \(y\)-coordinate of any other point on the line.
Instead of evaluating the formula to find \(m\text{,}\) we substitute the values we know for \(m\) and for \((x_1, y_1)\text{.}\) If we then plug in the \(x\)-coordinate of any unknown point, we can solve for \(y\text{.}\)
A line has slope \(\dfrac{-3}{4}\) and passes through the point \((1,-4)\text{.}\) Which equation can you use to find the \(y\)-coordinate of the point on the line with \(x\)-coordinate 6?
Remember that the equation for a line is really just a formula that gives the \(y\)-coordinate of any point on the line in terms of its \(x\)-coordinate. So, if we know the slope of a particular line and one point on the line, we can use the coordinate formula for slope to find its equation.
We first plot the given point, \((1,-4)\text{,}\) and then use the slope to find another point on the line. The slope is \(m =\dfrac{-3}{4}=\dfrac{\Delta y}{\Delta x} \text{,}\) so starting from \((1,-4)\) we move down 3 units and then 4 units to the right. This brings us to the point \((5,-7)\text{.}\) We draw the line through these two points.
We substitute \(\dfrac{-3}{4} \) for the slope, \(m\text{,}\) and \((1,-4)\) for \((x_1,y_1)\text{.}\) For the second point, \((x_2,y_2)\text{,}\) we substitute the variable point \((x,y) \) to obtain
When we use the slope formula to find the equation of a line, we substitute a variable point \((x, y)\) for the second point. This version of the formula,
\begin{equation*}
m = \frac{y - y_1}{x - x_1}
\end{equation*}
is called the point-slope form for a linear equation. It is sometimes stated in another version by clearing the fraction to get
\begin{align*}
\alert{(x-x_1)}m\amp = \frac{y-y_1}{x-x_1}\alert{(x-x_1)} \\
(x-x_1)m \amp = y - y_1\\
y \amp = y_1 + m(x-x_1)
\end{align*}
Now we are ready to find the equation promised in the introduction to this section: a formula for the freezing temperature of water in terms of its salinity.
Sea water does not freeze at exactly 32\(\degree\)F because of its salinity. The temperature at which water freezes depends on its dissolved mineral content. A common unit for measuring salinity is parts per thousand, or ppt. For example, salinity of 8 ppt means 8 grams of dissolved salts in each kilogram of water. Here are some data for the freezing temperature of water.
The fee for registering a new car is given by a linear equation that depends on the carβs value. The fee for a $15,000 car is $128.50, and the fee for a $25,000 car is $193.50.
In this section, we studied three different formulas associated with linear equations: the slope-intercept formula, the coordinate formula for slope, and the point-slope formula. How are these formulas related, and how are they different?
The slope-intercept form, \(y = mx + b\text{,}\) is just a special case of the point-slope formula. If the given point \((x_1, y_1)\) happens to be the \(y\)-intercept \((0, b)\text{,}\) then the point-slope formula reduces to the familiar form:
\begin{align*}
y\amp =y_1+m(x-x_1) \amp\amp\blert{\text{Substitute } b \text{ for } y_1 \text{ and 0 for } x_1.}\\
y\amp = b + m(x-0) \amp\amp \blert{\text{Simplify.}}\\
y\amp = mx + b
\end{align*}
We can use the (shorter) slope-intercept form if we are lucky enough to know the \(y\)-intercept of the line.
They are really the same formula, but they are used for different purposes:
The slope formula is used to calculate the slope when we know two points. We know \((x_1, y_1)\) and \((x_2, y_2)\text{,}\) and we are looking for \(m\text{.}\)
The boiling point of water changes with altitude and is approximated by the formula
\begin{equation*}
B = 212 - 0.0018h
\end{equation*}
where \(B\) is in degrees and \(h\) is in feet. State the slope and vertical intercept of the graph, including units, and explain their meaning in this context.
Flying lessons cost $645 for an 8-hour course and $1425 for a 20-hour course. Both prices include a fixed insurance fee. Write an equation for the cost, \(C\text{,}\) of flying lessons in terms of the length, \(h\text{,}\) of the course in hours.
In the desert, the sun rose at 6 am. At 9 am the temperature was \(80\degree, \) and at 3 pm the temperature was \(110\degree. \) Write an equation for the temperature \(T, \) at \(h\) hours after sunrise.
A radio station in Detroit, Michigan, reports the high and low temperatures in the Detroit/Windsor area as \(59\degree\)F and \(23\degree\)F, respectively. A station in Windsor, Ontario, reports the same temperatures as \(15\degree\)C and \(-5\degree\)C. Express the Fahrenheit temperature, \(F\text{,}\) in terms of the Celsius temperature, \(C\text{.}\)
The gas tank in Cicelyβs car holds \(14\) gallons. When the tank was half full her odometer read \(308\) miles, and when she filled her tank with \(12\) gallons of gasoline the odometer read \(448\text{.}\) Express her odometer reading, \(m, \) in terms of the amount of gas, \(g, \) she used.