Chapter 1.7, Problem 117E

Stopping Distance For a certain model of car the distance d required to stop the vehicle if it is traveling at v mi/h is given by the formula

$d=v+\frac{v^2}{20}$

where d is measured in feet. Kerry wants her stopping distance not to exceed 240 ft. At what range of speeds can she travel?

To determine

The range of speed for which the vehicle has stopping distance that do not exceed 240ft.

Answer to Problem 117E

The range of speed is $\underset{\_}{\text{between 0 mi/h and 60 mi/h}}$.

Explanation of Solution

The formula for distance to stop the car is $d=v+\frac{v^2}{20}$ with the stopping distance 240ft or less, where d is the distance measured in feet (ft) and v is the speed of the car measured in mi/h.

Substitute the value of 240ft for d in $d=v+\frac{v^2}{20}$ and calculate the value of v as follows.

Since the stopping distance is 240ft or less, the formula $d=v+\frac{v^2}{20}$ will become $240\ge v+\frac{v^2}{20}$. That is,

$\begin{array}{l}240\ge v+\frac{v^2}{20}\\ v+\frac{v^2}{20}-240\le 0\\ v^2+20v-4800\le 0\end{array}$

On further simplification,

$\begin{array}{l}{v}^2+20v-4800\le 0\\ v^2+\left(80-60\right)v-4800\le 0\\ v^2+80v-60v-4800\le 0\end{array}$

Further solve the equation as follows.

$\begin{array}{l}v\left(v+80\right)-60\left(v+80\right)\le 0\\ \left(v+80\right)\left(v-60\right)\le 0\end{array}$

Thus, $v\le -80$ and $v\le 60$.

Since the speed cannot be negative, $0\le v\le 60$.

Therefore, the range of speed is $\underset{\_}{\text{between 0 mi/h and 60 mi/h}}$.