The introduction to Chapter 1 discussed the rise in sea level over the last few decades. One of the contributing factors to this rise is the melting of Earthβs glaciers.
Glaciers around the world are retreating at accelerating rates, and many lower-latitude mountain glaciers may soon disappear entirely. The meltwater from these smaller glaciers contributed as much as 40% to the total rise in sea level over the 1990s.
By measuring the change in ice and snow height at fixed points and multiplying by the surface area of the glacier, scientists calculate the total volume of water lost from land-based glaciers. Dividing this volume by the surface area of the worldβs oceans gives the resulting change in sea level.
The graphs below show the change in thickness of the land-based glaciers overthe past 5 years, and the rise in sea level attributed to their melting. In this section we consider how to measure a rate of change.
The equation we use to describe a linear model, \(y = mx + b\text{,}\) gives the starting value \(b\) of the variable \(y\) and the rate \(m\) at which \(y\) changes. Now we will look more closely at rates and how they appear on the graph of the model. First, letβs review the notion of a ratio used as a rate. You are already familiar with several types of rates.
In the examples below, notice that each rate has units of the form \(\dfrac{\text{something}}{\text{something else}} \text{,}\) which we read as "something per something else."
The rate you calculated in Practice 1, Delbertβs average speed, actually compares the change in two variables, his distance from his starting point, and the time elapsed. This type of rate appears often in linear models.
Gregor is driving across Montana. At 1 pm his trip odometer reads 189 miles, and at 4 pm it reads 360 miles. Calculate Gregorβs average speed as a rate of change.
Gregorβs speed is the ratio of the distance he traveled to the time it took. The distance he traveled is the change in his odometer reading (from 189 miles to 360 miles), and the time it took is the change in the clock reading (from 1 pm to 4 pm).The units of this ratio are miles per hour.
How do we see the rate of change on a graph? Letβs consider the example above. The graph shows Gregorβs distance, \(d\text{,}\) at time \(t\text{.}\) We plot the two data points, \((1, 189)\) and \((4, 360)\text{,}\) and draw a straight line joining them. We can illustrate \(\Delta d\) and \(\Delta t\) by vertical and horizontal line segments, as shown on the graph.
The rate of change of distance with respect to time, or speed, is the ratio of \(\Delta d\) to \(\Delta t\text{.}\) It measures how much \(d\) changes for each unit increase in \(t\text{,}\) or how far Gregor travels in each hour. This quantity, the ratio \(\dfrac{\Delta d}{\Delta t} \text{,}\) is the slope of the line. \(~\alert{\text{[TK]}}\)
The slope tells us how much the \(y\)-coordinate changes for each unit of increase in the \(x\)-coordinate, as we move from one point to another along the line.
\begin{equation*}
\Delta x~ \text{ is}~
\begin{cases}
\text{positive if }x\text{ increases} \amp \text{(we move to the right)}\\
\text{negative if }x\text{ decreases} \amp \text{(we move to the left)}
\end{cases}
\end{equation*}
For positive slopes, the larger the value of \(m\text{,}\) the more the \(y\)-value increases for each unit increase in \(x\text{,}\) and the more we climb up as our location changes from left to right. (So graph A is steeper than graph B.)
If \(y\) decreases as we move from left to right, then \(\Delta y\) is negative when \(\Delta x\) is positive, so their ratio (the slope) is negative. (See graph C.)
The points \((20,500)\) and \((40,1000) \) lie on the graph. As we move from the first point to the second point, \(x\) increases by 20 units, so \(\Delta x=20\text{,}\) and \(y\) increases by 500 units, so \(\Delta y = 500\text{.}\) Thus
The slope is 25, which means that \(y\) increases 25 units for each 1-unit increase in \(x\text{.}\) So, if \(\Delta x=1\text{,}\) then \(\Delta y=25\text{.}\)
By putting together all of our discussion so far, we can now see that the slope of a line measures the rate of change of \(y\) with respect to \(x\text{.}\)
\begin{equation*}
y = (\text{starting value}) + (\text{rate}) \times t
\end{equation*}
From the graph, we see that at \(t = 0\) the skierβs altitude was \(a = 100\text{,}\) and we calculated the rate of change of altitude as 150 feet per minute. Substituting these values into the formula, we find
For the line shown in the figure, try computing the slope using the points \(P\) and \(Q\text{,}\) and then using the points \(G\) and \(H\text{.}\) In each case, you should find that the slope is \(\frac{3}{2} \text{.}\)
Here is another way to look at slopes. If we start at any point on the line shown and move 9 units to the right, what value of \(\Delta y\) will bring us back to the line? We can use the slope formula with \(\Delta x=9\text{.}\)
The fact that lines have constant slope has two important consequences. First because \(m\) is constant for a given line, we can use the formula \(m=\dfrac{\Delta y}{\Delta x} \)to find \(\Delta y\) when we know \(\Delta x \text{,}\) or to find \(\Delta x \) when we know \(\Delta y \text{.}\)
We first draw a sketch of the wheelchair ramp and label \(\Delta x\) and \(\Delta y\text{.}\) We are given that \(\Delta y=4\) feet, and we are looking for \(\Delta x\text{.}\) We substitute the known values into the slope formula, and solve for \(\Delta x\text{.}\)
Here is a second consequence of the fact that lines have constant slope: We can tell whether a collection of data points lies on a straight line by computing slopes. If the slopes between pairs of data points are all the same, the points lie on a straight line.
If we know the slope of a line, we can use the formula \(m=\dfrac{\Delta y}{\Delta x} \)to find \(\Delta y\) when we know \(\Delta x \text{,}\) or to find \(\Delta x \) when we know \(\Delta y \text{.}\)
To test whether a collection of data points lies on a straight line, we can compute slopes. If the slopes between all pairs of data points are the same, the points lie on a straight line.
On his 512-mile round trip to Las Vegas and back, Corey needed 16 gallons of gasoline. He used 13 gallons of gasoline on a 429-mile trip to Los Angeles. On which trip did he get better fuel economy?
Grimy Gulch Pass rises 0.6 miles over a horizontal distance of 26 miles. Bobβs driveway rises 12 feet over a horizontal distance of 150 feet. Which is steeper?
Which is steeper, the truck ramp for Acme Movers, which rises 4 feet over a horizontal distance of 9 feet, or a toy truck ramp, which rises 3 centimeters over a horizontal distance of 7 centimeters?
Start at point \((0,2)\) and move 4 units in the positive \(x\)-direction. How many units must you move in the \(y\)-direction to get back to the line? What is the ratio of \(\Delta y\) to \(\Delta x\text{?}\)
Start at point \((0,2)\) and move \(-6\) units in the positive \(x\)-direction. How many units must you move in the \(y\)-direction to get back to the line? What is the ratio of \(\Delta y\) to \(\Delta x\text{?}\)
Suppose you start at any point on the line and move 18 units in the \(x\)-direction. How many units must you move in the \(y\)-direction to get back to the line? Use the equation \(m=\dfrac{\Delta y}{\Delta x}\) to calculate your answer.
Start at point \((0,-6)\) and move \(-6\) units in the \(y\)-direction (down). How many units must you move in the \(x\)-direction to get back to the line? What is the ratio of \(\Delta y\) to \(\Delta x\text{?}\)
Start at point \((0,-6)\) and move 9 units in the positive \(y\)-direction. How many units must you move in the \(x\)-direction to get back to the line? What is the ratio of \(\Delta y\) to \(\Delta x\text{?}\)
Suppose you start at any point on the line and move 24 units in the \(y\)-direction. How many units must you move in the \(x\)-direction to get back to the line? Use the equation \(m=\dfrac{\Delta y}{\Delta x}\) to calculate your answer.
Residential staircaises are usually built with a slope of 70%, or \(\dfrac{7}{10}\text{.}\) If the vertical distance between stories is 10 feet, how much horizontal space does the staircase require?
\begin{align*}
\amp\text{change in population}=\Delta P = \\
\amp\text{time elapsed}=\Delta t = \\
\amp\text{rate of growth}= \frac{\Delta P}{\Delta t} =
\end{align*}
A traditional first experiment for chemistry students is to make 98 observations about a burning candle. Delbert records the height, \(h\text{,}\) of the candle in inches at various times \(t\) minutes after he lit it.
A spring is suspended from the ceiling. The table shows the length of the spring, in centimeters, as it is stretched by hanging various weights from it.
Geologists calculate the speed of seismic waves by plotting the travel times for waves to reach seismometers at known distances from the epicenter. The speed of the wave can help them determine the nature of the material it passes through. The graph shows a travel time graph for P-waves from a shallow earthquake.
Niagara Falls was discovered by Father Louis Hennepin in 1682. In 1952, much of the water of the Niagara River was diverted for hydroelectric power, but until that time erosion caused the Falls to recede upstream by 3 feet per year.
The Falls were formed about 12,000 years ago during the end of the last ice age. How far downstream from their current position were they then? (Give your answer in miles.)
Naismithβs Rule is used by runners and walkers to estimate journey times in hilly terrain. In 1892 Naismith wrote in the Scottish Mountaineering Club Journal that a person βin fair condition should allow for easy expeditions an hour for every three miles on the map, with an additional hour for every 2000 feet of ascent.β (Source: Scarf, 1998.)
According to Naismith, 1 unit of ascent requires the same travel time as how many units of horizontall travel? (This is called Naismithβs number.) Round your answer to one decimal place.
The graph appears to be almost linear from 1992 to 2002. Read the graph to complete the table, then compute the slope of the graph over that time interval, including units. What does the slope mean in this situation?
From 1960 to 2003, the land-based glaciers decreased in thickness by about 8 meters (or 0.008 km). The total area of those glaciers is 785,000 square kilometers. Calculate the total volume of water released by melting. (Hint: \(\text{Volume}= \text{area}\times \text{thickness} \))
The surface area of the worldβs oceans is 361.6 million square kilometers. When the meltwater from the land-based glaciers (thatβs the volume you calculated in part (d)) enters the oceans, how much will the sea level rise, in kilometers? Use the formula in part (d). Convert your answer to millimeters, and check your answer against your answer to part (b).