Chapter 9.3, Problem 30HP

a.

To determine

To explain: Relation between steepness of the line and rate of change.

Answer to Problem 30HP

Greater the steepness of a line, greater is the rate of change.

Explanation of Solution

Given information:

The following graph:

Greater the steepness of a line, greater is the rate of change. Steeper lines represent a greater transition in the y- axis for every x- coordinate as compared to less-steeper lines. In the given graph, the line representing mosquitoes is steeper than the line representing honey bees. Hence, the number of beats per second of a mosquito’s wing is greater than the number of beats per second of a honey bee’s wing.

For graph of mosquito, rate of change is calculated as:

  $\begin{array}{l}\text{Rate of change = }\frac{\text{Change in number of beats }}{\text{Change in time}}\\ =\frac{\left(600-300\right)\text{ beats}}{\left(1-0.5\right)\text{ h}}[\text{Substitute]}\\ =\frac{300\text{ beats}}{0.5\text{ h}}[\text{Simplify]}\\ =\frac{\text{600 beats}}{1\text{ h}}[\text{Write as unit rate]}\end{array}$

For graph of honey bee, rate of change is calculated as:

  $\begin{array}{l}\text{Rate of change = }\frac{\text{Change in number of beats }}{\text{Change in time}}\\ =\frac{\left(400-200\right)\text{ beats}}{\left(2-1\right)\text{ h}}[\text{Substitute]}\\ =\frac{200\text{ beats}}{\text{1 h}}[\text{Simplify]}\end{array}$

b.

To determine

To explain: Relation between unit rate and slope.

Answer to Problem 30HP

Rate of change and slope are equal.

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:

For graph of mosquito,

Rate of change is calculated as:

  $\begin{array}{l}\text{Rate of change = }\frac{\text{Change in number of beats }}{\text{Change in time}}\\ =\frac{\left(600-300\right)\text{ beats}}{\left(1-0.5\right)\text{ h}}[\text{Substitute]}\\ =\frac{300\text{ beats}}{0.5\text{ h}}[\text{Simplify]}\\ =\frac{\text{600 beats}}{1\text{ h}}[\text{Write as unit rate]}\end{array}$

Slope is calculated as:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x}[\text{Definition of slope]}\\ m=\frac{600-300}{1-0.5}[(x_1,y_1)=(0.5,300),(x_2,y_2)=(1,600)]\\ m=\frac{300}{0.5}[\text{Simplify]}\\ m=600[\text{Divide]}\end{array}$

For graph of honey bee,

Rate of change is calculated as:

  $\begin{array}{l}\text{Rate of change = }\frac{\text{Change in number of beats }}{\text{Change in time}}\\ =\frac{\left(400-200\right)\text{ beats}}{\left(2-1\right)\text{ h}}[\text{Substitute]}\\ =\frac{200\text{ beats}}{\text{1 h}}[\text{Simplify]}\end{array}$

Slope is calculated as:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x}[\text{Definition of slope]}\\ m=\frac{400-200}{2-1}[(x_1,y_1)=(1,200),(x_2,y_2)=(2,400)]\\ m=\frac{200}{1}[\text{Simplify]}\\ m=200[\text{Divide]}\end{array}$

For both the graphs, rate of change and slope are equal. Except that the rate of change has a unit while a slope does not.