Chapter 4.7, Problem 60HP

(a)

To determine

To find:The slope of the uphill and downhill portion of the trip.

Answer to Problem 60HP

The slope of the uphill trip is ¼ and the slope of the downhill trip is -(1/4).

Explanation of Solution

Given:

The vertical cross-section of the hill is modelled by the equation

  $y=-\frac{1}{4}\left|x-400\right|+100$

Calculation:

Plot the function of the hill.

  

For the range $0\le x\le 800$ ,

Calculate the slope of the uphill of the trip:

  $\begin{array}{c}{m}_1=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{100-0}{400-0}\\ =\frac{1}{4}\end{array}$

Calculate the slope of the downhill trip:

  $\begin{array}{c}{m}_2=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{0-100}{800-400}\\ =-\frac{1}{4}\end{array}$

Therefore, the slope of the uphill trip is ¼ and the slope of the downhill trip is (-1/4).

(b)

To determine

To plot:The graph of the function.

Answer to Problem 60HP

The graph of the function is drawn below.

Explanation of Solution

Given:

The vertical cross-section of the hill is modelled by the equation

  $y=-\frac{1}{4}\left|x-400\right|+100$

Calculation:

In the given function, a vertical shift of 100 is observed from the equation. Since the function is negative, the graph of the function opens in the downward direction.

Plot the function of the hill.

  

Therefore, the graph of the function is drawn above.

(c)

To determine

To find:The domain and the range of the function of the hill.

Answer to Problem 60HP

The domain of the function is $-\infty <x<\infty$ and the range of the function is $f\left(x\right)\le 100\text{ or }[-\infty ,100)$ .

Explanation of Solution

Given:

The vertical cross-section of the hill is modelled by the equation

  $y=-\frac{1}{4}\left|x-400\right|+100$

Calculation:

Plot the function of the hill.

  

For the range $0\le x\le 800$ ,

The set of input or argument values for which the function gives real and defined output is termed as the domain of the function. Find the domain from the given graph. In the given graph domain constraints and the undefined points are not observed. Therefore the domain of the graph is $-\infty <x<\infty$ .

The range is defined as the set of values of the dependent variables for which the function has defined values.

The range of the given function is $f\left(x\right)\le 100\text{ or }[-\infty ,100)$ .

Therefore the domain of the graph is $-\infty <x<\infty$ and the range of the graph is $f\left(x\right)\le 100\text{ or }[-\infty ,100)$ .