Question

A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch?

Note: 1 mile = 5280 feet

$\dfrac{\pi}{11}$

$\dfrac{\pi}{10}$

$\dfrac{\pi}{5}$

$\dfrac{2\pi}{5}$

$\dfrac{2\pi}{3}$

Solution:

Each semicircle advances Robert $40$ feet along the highway, so a one-mile stretch requires $5280/40=132$ semicircles.

Each semicircle has diameter $40$ feet, so its arc length is $20\pi$ feet. The total distance Robert rides is $132\cdot20\pi=2640\pi$ feet, which is $\dfrac{2640\pi}{5280}=\dfrac{\pi}{2}$ miles.

At $5$ miles per hour, the time is $\dfrac{\pi}{2}\div5=\dfrac{\pi}{10}$ hours.

Thus, B is the correct answer.

2014 AMC 8 Problem 25

Solution:

Problem 25 in Other Years