Chapter 1.8, Problem 55E

(a)

To determine

To find: The function that represents the total stopping distance $T$ .

Answer to Problem 55E

  $\frac{3}{4}x+\frac{1}{15}{x}^2$ .

Explanation of Solution

Given information:

The research and development department of an automobile manufacturer determines that when a driver is required to stop quickly to avoid an accident, the distance the car travels during the driver’s reaction time is given by $R\left(x\right)=\frac{3}{4}x$ , where $x$ is the speed of the car in miles per hour. The distance the car travels while the driver is braking is given by $B\left(x\right)=\frac{1}{15}{x}^2$ .

Calculation:

The total stopping distance $T\left(x\right)$ is the sum of the two functions $R\left(x\right)$ and $B\left(x\right)$

  $\begin{array}{c}T\left(x\right)=R\left(x\right)+B\left(x\right)\\ =\frac{3}{4}x+\frac{1}{15}{x}^2\end{array}$

(b)

To determine

To sketch: the graph the functions $R,B$ and $T$ on the same set of coordinate axes for $0\le x\le 60$ .

Explanation of Solution

Calculation:

The functions $R\left(x\right),B\left(x\right)$ and $T\left(x\right)$ are plotted below.

  

(c)

To determine

To find: the function contributes most to the magnitude of the sum at higher speeds, explain.

Answer to Problem 55E

The braking function $B\left(x\right)$ . As $x$ increases, $B\left(x\right)$ increases at a faster rate than $R\left(x\right)$ .

Explanation of Solution

The braking function $B\left(x\right)$ . As $x$ increases, $B\left(x\right)$ increases at a faster rate than $R\left(x\right)$ .