Q. 107 - Precalculus Enhanced with Graphing Utilities

Question

Two pulleys, one with radius 2 inches and the other with radius 8 inches, are connected by a belt. (See the figure.) If the 2-inch pulley is caused to rotate at 3 revolutions per minute, determine the revolutions per minute of the 8-inch pulley. 

Step-by-Step Solution

Answer

The revolution per minute of 8-inch pulley is 0.75 rpm.

Step 1: Given

Radius of small pulley$\left(r_1\right)=2 \mathrm{inch}$

Radius of large pulley$\left(r_2\right)=8 \mathrm{inch}$

The 2-inch pulley is caused to rotate at 3 revolutions per minute.

Step 2: Calculate the revolution per minute of large pulley.

The circumference of the small pulley is computed as,

$C_1 = 2{\mathrm{\pi r}}_1=2\times 3.14\times 2=12.56 \mathrm{inch}$

The circumference of the large pulley is computed as,

$C_2=2{\mathrm{\pi r}}_2=2\times 3.14\times 8=50.24 \mathrm{inch}$

Since the revolution per minute of the pulley is inversely proportional to its radius.

Therefore,

$$\begin{gathered} \frac{\mathrm{Rpm} \mathrm{of} \mathrm{small} \mathrm{pully}}{\mathrm{Rpm} \mathrm{of} \mathrm{large} \mathrm{pulley}}=\frac{{\mathrm{C}}_2}{{\mathrm{C}}_1} \\ \frac{3}{\mathrm{Rpm} \mathrm{of} \mathrm{large} \mathrm{pulley}}=\frac{50.24}{12.56} \\ \mathrm{Rpm} \mathrm{of} \mathrm{large} \mathrm{pulley}=3\times \frac{12.56}{50.24} \\ \mathrm{Rpm} \mathrm{of} \mathrm{large} \mathrm{pulley}=0.75 \end{gathered}$$

The revolution per minute of the 8-inch pulley is 0.75 rpm.