Problem 252

Solution 1
There are two possible interpretations of the problem: that the road as a whole is $40$ feet wide, or that each lane is $40$ feet wide. Both interpretations will arrive at the same result. However, let us stick with the first interpretation for simplicity. Each lane must then be $20$ feet wide, so Robert must be riding his bike in semicircles with radius $20$ feet and diameter $40$ feet. Since the road is $5280$ feet long, over the whole mile, Robert rides $\frac{5280}{40} =132$ semicircles in total. Were the semicircles full circles, their circumference would be $2\pi\cdot 20=40\pi$ feet; as it is, the circumference of each is half that, or $20\pi$ feet. Therefore, over the stretch of highway, Robert rides a total of $132\cdot 20\pi =2640\pi$ feet, equivalent to $\frac{\pi}{2}$ miles. Robert rides at 5 miles per hour, so divide the $\frac{\pi}{2}$ miles by $5$ mph (because $t = \frac{d}{r}$ and time = distance/rate) to arrive at $\boxed{\textbf{(B) }\frac{\pi}{10}}$ hours.

Solution 2
If Robert rides in a straight line, it will take him $\frac{1}{5}$ hours. When riding in semicircles, let the radius of the semicircle $r$, the circumference of a semicircle is $\pi r$. The ratio of the circumference of the semicircle to its diameter is $\frac{\pi}{2}$, so the time Robert takes is $\frac{1}{5} \cdot \frac{\pi}{2}$, which equals to $\boxed{\textbf{(B) }\frac{\pi}{10}}$ hours.