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
.
See also
2000 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |