8-1Q A bicycle odometer (which counts... [FREE SOLUTION]

Chapter 8: 8-1Q (page 198)

A bicycle odometer (which counts revolutions and is calibrated to report distance traveled) is attached near the wheel axle and is calibrated for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Short Answer

Expert verified

The rotations of the smaller wheel of the bicycle will appear on the odometer as 9.43 inches more than the actual distance moved by the bicycle.

Step by step solution

Identification of given data

The given data can be listed below as:

The diameter of the wheels of one bicycle is\({d_1} = 27{\rm{ inches}}\).
The diameter of the wheels of another bicycle is\({d_2} = 24{\rm{ inches}}\).

Understanding the measurement of the distance on the odometer of the bicycle

The odometer measures the particular distance traveled by the bicycle. An odometer counts the revolutions of the bicycle wheel. One rotation of the wheel is equal to the value of the circumference of the bicycle. The circumference of the bicycle is equal to\(\left( \pi \right)\)times the diameter of the bicycle.

Determination of the distance moved by the bicycle

The distance traveled by the bicycle can be expressed as:

\({R_1} = \pi {d_1}\)

Here,\({d_1}\)is the diameter of the bicycle wheel.

Substitute the values in the above equation.

\(\begin{aligned}{c}{R_1} = \pi \times 27{\rm{ inches}}\\ = 84.82{\rm{ inches}}\end{aligned}\)

The distance traveled by another bicycle can be expressed as:

\({R_2} = \pi {d_2}\)

Here,\({d_2}\)is the diameter of another bicycle wheel.

Substitute the values in the above equation.

\(\begin{aligned}{c}{R_2} = \pi \times 24{\rm{ inches}}\\ = 75.39{\rm{ inches}}\end{aligned}\)

Determination of the extra distance moved by the 27 inches wheel

The extra distance traveled by the 27 inches wheel of the bicycle can be given as:

\(R = {R_1} - {R_2}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}R = 84.82{\rm{ inches}} - 75.39{\rm{ inches}}\\ = 9.43{\rm{ inches}}\end{aligned}\)

Thus, the rotations of the smaller wheel of the bicycle will appear on the odometer as 9.43 inches more than the actual distance moved by the bicycle.