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 miles per hour along the highways and at 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 square miles, where and are relatively prime positive integers. Find .

Solution

Place the intersection of the highways at the origin and let the highways be the x and y axis. We consider the case where the truck moves in +x. After going x miles, hours has passed. If the truck leaves the highway it can travel for at most hours, or miles. It can end up anywhere off the highway in a circle with this radius centered at . All these circle are homothetic with center at . Now consider the circle at (0,0). Draw a line tangent to it at and passing through . By the Pythagorean Theorem so . . The slope of line is therefore . Since it passes through its equation is . The line and the x and y axis bound the region the truck can go if it moves in +x. Similarly, the line bounds the region the truck can go if it moves in +y. The intersection of these 2 lines is . The bounded region in Quadrant I is made up of a square and 2 triangles. . By symmetry, the regions in the other quadrants are the same, so the area of the whole region is so the answer is .