Chapter 8.4, Problem 15HP

To determine

To find:The connection between the steepness of the lines and the rates of change.

Answer to Problem 15HP

The steeper the line, the greater the rate of change.

Explanation of Solution

Given:

Consider an example and explain

Calculation:

Consider the table that shows time and number of beats taken by mosquitoes:

Time (s) [x]	Number of beats [y]
0.5	300
1	600

And also, make the table that shows time and number of beats taken by honey bees:

Time (s) [x]	Number of beats [y]
1	200
2	400

The rate of change is change in one quality with respect to another quantity.

Mathematically:

  $Rateof\text{}\text{}\text{}\text{}change=\frac{change\text{}\text{}iny}{changeinx}$

Thus, the rate of change is change in number of beats with respect to change in time.

So, the rate of change for mosquitoes is:

  $\begin{array}{l}\text{Rate of change=}\frac{\text{600-300}}{\text{1-0}\text{.5}}\\ \text{=}\frac{\text{300}}{\text{0}\text{.5}}\\ \text{= 600}\end{array}$

And, the rate of change for honey bees is:

  $\begin{array}{l}\text{Rate of change=}\frac{\text{400-200}}{\text{2-1}}\\ \text{= }\frac{\text{200}}{\text{1}}\text{=200}\end{array}$

The graph shows that the line that shows the speed at which mosquitoes beat their wing is steeper as compare to the speed at which honey bees beat their wings.

And also, the rate of change for mosquitoes is higher as the rate of change for honey bees.

Conclusion:

Hence, the steeper the line, the greater the rate of change.