Difference between revisions of "2014 AMC 8 Problems/Problem 25"
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==Solution 2: == | ==Solution 2: == | ||
If Robert rides in a straight line, it will take him <math>\frac{1}{5}</math> hours. When riding in semicircles, let the radius of the semicircle <math>r</math>, then the circumference of a semicircle is <math>\pi r</math>. The ratio of the circumference of the semicircle to its diameter is <math>\frac{\pi}{2}</math>, so the time Robert takes is <math>\frac{1}{5} * \frac{\pi}{2}</math>, which equals to <math>\boxed{\textbf{(B) }\frac{\pi}{10}}</math> hours. | If Robert rides in a straight line, it will take him <math>\frac{1}{5}</math> hours. When riding in semicircles, let the radius of the semicircle <math>r</math>, then the circumference of a semicircle is <math>\pi r</math>. The ratio of the circumference of the semicircle to its diameter is <math>\frac{\pi}{2}</math>, so the time Robert takes is <math>\frac{1}{5} * \frac{\pi}{2}</math>, which equals to <math>\boxed{\textbf{(B) }\frac{\pi}{10}}</math> hours. | ||
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Revision as of 02:47, 3 November 2019
Problem
A straight one-mile stretch of highway, feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at
miles per hour, how many hours will it take to cover the one-mile stretch?
Note: mile =
feet
Solution 1:
There are two possible interpretations of the problem: that the road as a whole is feet wide, or that each lane is
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
feet wide, so Robert must be riding his bike in semicircles with radius
feet and diameter
feet. Since the road is
feet long, over the whole mile, Robert rides
semicircles in total. Were the semicircles full circles, their circumference would be
feet; as it is, the circumference of each is half that, or
feet. Therefore, over the stretch of highway, Robert rides a total of
feet, equivalent to
miles. Robert rides at 5 miles per hour, so divide the
miles by
mph (because t = d/r time = distance/rate) to arrive at
hours.
Solution 2:
If Robert rides in a straight line, it will take him hours. When riding in semicircles, let the radius of the semicircle
, then the circumference of a semicircle is
. The ratio of the circumference of the semicircle to its diameter is
, so the time Robert takes is
, which equals to
hours.