Chapter 8.4, Problem 18HP

To determine

To explain:how rate of change affects the graph of the airplane’s position.

Answer to Problem 18HP

Thus, if rate of change is positive, then graph is increasing.

Explanation of Solution

Calculation:

Consider the table that shows vertical and horizontal position of airplane at different points.

Points	Horizontal position (in feet)	Vertical position (in feet)
A	0	2000
B	400	2000
C	800	2400
D	1100	2000
E	1400	2000
F	1600	2800

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

Mathematically,

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

So, rate of change from point A to B is:

  $\begin{array}{l}\text{Rate of change=}\frac{\text{2000-2000}}{\text{400-0}}\\ \text{=}\frac{\text{0}}{\text{400}}\text{=0}\end{array}$

Thus, if vertical position remains same, then rate of change is constant and graph is horizontal.

Now, rate of change from point B to C is:

  $\begin{array}{l}Rateofchang=\frac{2400-2000}{800-400}\\ =\frac{400}{400}=1\end{array}$

Thus, if rate of change is positive, then graph is increasing.

Rate of change from point C to D is:

  $\begin{array}{l}\text{Rate of change=}\frac{\text{2000-2400}}{\text{1100-800}}\\ \text{=}\frac{\text{-400}}{\text{300}}\text{=}\frac{\text{-4}}{\text{3}}\end{array}$

Thus, if rate of change is negative, then graph is decreasing.

Rate of change from point D to E is:

  $\begin{array}{l}\text{Rate of change=}\frac{\text{2000-2000}}{\text{1400-1100}}\\ \text{=}\frac{\text{0}}{\text{300}}\text{=0}\end{array}$

Thus if vertical position remains same, then rate of change is constant and graph is horizontal.

Now, rate of change from point E to F is:

  $\begin{array}{l}\text{Rate of change=}\frac{\text{2800-2000}}{\text{1600-1400}}\\ \text{=}\frac{\text{800}}{\text{200}}\text{=4}\end{array}$

Thus, if rate of change is positive, then graph is increasing.

Conclusion:

Therefore, if rate of change is positive, then graph is increasing.