2000 AIME I Problems/Problem 13

Problem

In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is $m/n$ square miles, where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Solution

Place the intersection of the highways at the origin $O$ and let the highways be the x and y axis. We consider the case where the truck moves in +x. After going x miles, $t=\frac{d}{r}=\frac{x}{50}$ hours has passed. If the truck leaves the highway it can travel for at most $t=\frac{1}{10}-\frac{x}{50}$ hours, or $d=rt=14t=1.4-\frac{7x}{25}$ miles. It can end up anywhere off the highway in a circle with this radius centered at $(x,0)$ . All these circle are homothetic with center at $(5,0)$ . Now consider the circle at (0,0). Draw a line tangent to it at $A$ and passing through $B (5,0)$ . By the Pythagorean Theorem $AB^2+AO^2=OB^2$ so $AB=\sqrt{OB^2-AO^2}=\sqrt{5^2-1.4^2}=\frac{24}{5}$ . $tan(<ABO)=frac{OA}{AB}=\frac{7}{24}$ . The slope of line $AB$ is therefore $\frac{7}{24}$ .

The area of the region is not $\frac{700}{31}$ so the answer is $700+31=\boxed{731}$ .

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2000 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2000 AIME I Problems/Problem 13

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