2000 AIME I Problems/Problem 13

From AoPSWiki

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

Let the intersection of the highways be at the origin , and let the highways be the x and y axes. We consider the case where the truck moves in the positive x direction.

After going 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 circles are homothetic with respect to a center at .

     

Now consider the circle at . Draw a line tangent to it at and passing through . By the Pythagorean Theorem . Then , so the slope of line is . Since it passes through its equation is .

This line and the x and y axis bound the region the truck can go if it moves in the positive x direction. Similarly, the line bounds the region the truck can go if it moves in positive y direction. The intersection of these two lines is . The bounded region in Quadrant I is made up of a square and two triangles. . By symmetry, the regions in the other quadrants are the same, so the area of the whole region is so the answer is .

See also