Q. 110 - Precalculus Enhanced with Graphing Utilities

Question

A spin balancer rotates the wheel of a car at 480 revolutions per minute. If the diameter of the wheel is 26 inches, what road speed is being tested? Express your answer in miles per hour. At how many revolutions per minute should the balancer be set to test a road speed of 80 miles per hour? 

Step-by-Step Solution

Answer

The road speed being tested is 37.13 miles per hour and the number of revolutions per minute should be 1034.26 to be set to test a road speed of 80 miles per hour.

Step 1: Given information

A spin balancer rotates the wheel of a car at 480 revolutions per minute and the diameter of the wheel is 26 inches.

Step 2: Calculate the road speed.

As we know,

$$\begin{gathered} \mathrm{Linear} \mathrm{velocity} \left(v\right)=r\times \frac{\theta }{t} \\ 1 \mathrm{revolution} = 2\mathrm{\pi } \mathrm{radian} \mathrm{per} \mathrm{minute} \end{gathered}$$

And,

$$\begin{gathered} 480 \mathrm{revolution}=480\times 2\mathrm{\pi } \\ = 960 \mathrm{radian} \mathrm{per} \mathrm{minute} \end{gathered}$$

So that,

 $v=13\times \frac{960\mathrm{\pi } \mathrm{radian}}{1 \mathrm{minute}}=39207.0763 \mathrm{inch} \mathrm{per} \mathrm{minute}$

1 mile$=63360 \mathrm{inch}$

1 hour$=60 \mathrm{minute}$

Therefore,

$$\begin{gathered} 39207.0763 \mathrm{inch}/\min =\frac{39207.0763}{63360}\times 60 \mathrm{mile}/\mathrm{h} \\ =37.13 \mathrm{mile}/\mathrm{h} \end{gathered}$$

Thus, the road speed being tested is 37.13 miles per hour.

Calculate the  revolutions per minute should the balancer be set to test a road speed of 80 miles per hour,

$\mathrm{Circumference}=2\mathrm{\pi r}=26\mathrm{\pi } \mathrm{inches}$

$\frac{80 \mathrm{miles}}{1 \mathrm{hour}}\times \frac{5280 \mathrm{ft}}{1 \mathrm{mile}}\times \frac{12 \mathrm{inch}}{1 \mathrm{ft}}\times \frac{1 \mathrm{rev}}{26\mathrm{\pi } \mathrm{inch}}\times \frac{1 \mathrm{hour}}{60 \mathrm{minute}}=1034.26 \mathrm{rev}./\mathrm{minute}$

The number of revolutions per minute should be 1034.26 to be set to test a road speed of 80 miles per hour.