Chapter 9.3, Problem 24IP

a.

To determine

To calculate: Constant rate of change for the given linear function. Also, interpret its meaning.

Answer to Problem 24IP

Constant rate of change for the given linear function is $-60\text{ miles per hour}$ .

Explanation of Solution

Given information:

The following graph:

Calculation:

Choose any two points from the line such as $\text{(1,240)}$ and $\text{(2,180)}$ .

The time changes from $1$ to $2$ and distance changes from $240$ to $180$ .

Find the unit rate to determine the constant rate of change.

  $\begin{array}{l}\text{ Rate of change = }\frac{\text{Change in distance}}{\text{Change in time}}\\ =\frac{(180-240)\text{ mi}\text{.}}{(2-1)\text{ h}}[\text{Substitute]}\\ =\frac{-\text{60 mi}\text{.}}{\text{1 h}}[\text{Simplify]}\end{array}$

The distance decreases by $\text{60 miles}$ every hour.

b.

To determine

To calculate: Slope of the line

Answer to Problem 24IP

Slope of the line is $-60$ .

Explanation of Solution

Given information:

The following graph:

Formula used:

Slope is given by the formula,

  $m=\frac{y_2-y_1}{x_2-x}$

where $x_2\ne x$ .

Calculation:

Take any two points from the line such as $\text{(1,240)}$ and $\text{(2,180)}$ .

Slope is calculated as follows:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x}[\text{Definition of slope]}\\ m=\frac{180-240}{2-1}[(x_1,y_1)=(1,240),(x_2,y_2)=(2,180)]\\ m=\frac{-60}{1}[\text{Simplify]}\\ m=-60[\text{Divide]}\end{array}$

c.

To determine

To compare: Slope of line and rate of change.

Answer to Problem 24IP

Slope of line and rate of change are equal.

Explanation of Solution

Given information:

The following graph:

The rate of change obtained in part (a) is $-60\text{ miles per hour}$ and the slope obtained in part (b) is $-60$ . The values of both are equal but a slope is written without a unit while the rate of change has a unit.

d.

To determine

To calculate: Time taken to drive from St. Louis to Chicago.

Answer to Problem 24IP

Time taken to drive from St. Louis to Chicago is $5\text{ hours}$ .

Explanation of Solution

Given information:

The following graph:

Calculation:

To find the time taken to drive from St. Louis to Chicago, use the rate of change.

According to the graph,

Distance between St. Louis and Chicago = $300\text{ miles}$

Rate of change = $-60\text{ miles per hour}$

Time taken to drive from St. Louis to Chicago is calculated as follows:

  $\begin{array}{l}-300\text{ mi}\times \frac{1\text{ h}}{-60\text{ mi}}[\text{Multiply with unit rate]}\\ =5\text{ h}\text{[ Simplify]}\end{array}$