Question

1. recreation suppose you can ride a bicycle a distance of 5 mi in 15 min. if you ride at a constant speed and if the bicycle’s wheels have diameter 27 in., find the wheels’ approximate rate of turn (in rpm).

Explanation:

Step1: Find the speed in inches per minute

First, convert the distance from miles to inches. We know that 1 mile = 63360 inches, so 5 miles is \(5\times63360 = 316800\) inches. The time is 15 minutes. So the speed \(v\) (in inches per minute) is \(\frac{316800}{15}=21120\) inches per minute.

Step2: Find the circumference of the wheel

The formula for the circumference \(C\) of a circle is \(C = \pi d\), where \(d\) is the diameter. Given \(d = 27\) inches, so \(C=\pi\times27\approx3.1416\times27 = 84.8232\) inches.

Step3: Calculate the rpm

The number of revolutions per minute (rpm) is the speed divided by the circumference. So rpm \(=\frac{21120}{84.8232}\approx249\) (rounded to a reasonable approximation).

Answer:

The wheels' approximate rate of turn is \(\boxed{249}\) rpm (the value may vary slightly depending on the approximation of \(\pi\) used, for example, if \(\pi\approx\frac{22}{7}\), the calculation would be a bit different but close to this result).