Chapter 1.7, Problem 39E

Solution

To Determine: The speed of the vehicle in miles per hour.

The speed of the vehicle is $11.42$ miles per hour.

Given:

Radius of the moving circle $r=16\text{in}$

Number of rotations per second $=2$

Calculation:

To find the speed, use the circumference of the circle formula:

  $C=2\pi r$

Substitute $r=16$ in the above formula:

  $\begin{array}{l}C=2\pi \times 16\\ C=32\pi \end{array}$

So, $32\pi$ inches are the distance that bicycle travels in $1$ rotation.

Tire making $2$ rotations per second:

  $32\pi \times 2=64\pi$

Thus, the distance travelled by the bicycle is $32\pi \times 2=64\pi$ inches per second.

  $\begin{array}{c}64\pi =64\times 3.14\\ =200.96\end{array}$

Thus, the distance travelled by the bicycle is $200.96$ inches per second.

To convert inches per seconds to miles per hour:

  $speed=\frac{distancetravelledbythevehicle}{numberofinchesinamile}\times Numberofsecondsinanhour$

Number of inches in a mile $=63360$

Number of seconds in an hour $=3600$

  $\begin{array}{c}v=\frac{200.96\times 3600}{63360}\\ v=\frac{723,456}{63360}\\ v=11.42\end{array}$

The speed of the vehicle is $11.42$ miles per hour.