1997 AIME Problems/Problem 7
Contents
Problem
A car travels due east at mile per minute on a long, straight road. At the same time, a circular storm, whose radius is miles, moves southeast at mile per minute. At time , the center of the storm is miles due north of the car. At time minutes, the car enters the storm circle, and at time minutes, the car leaves the storm circle. Find .
Solution 1
We set up a coordinate system, with the starting point of the car at the origin. At time , the car is at and the center of the storm is at . Using the distance formula,
Noting that is at the maximum point of the parabola, we can use .
Solution 2
First do the same process for assigning coordinates to the car. The car moves miles per minute to the right, so the position starting from is .
Take the storm as circle. Given southeast movement, split the vector into component, getting position for the storm's center. This circle with radius 51 yields .
Now substitute the car's coordinates into the circle's:
.
Simplifying and then squaring:
.
Forming into a quadratic we get the following, then set equal to 0, since the first time the car hits the circumference of the storm is and the second is .
The problem asks for sum of solutions divided by 2 so sum is equal to:
NOTE: This is literally the exact same as Solution 1...
See also
1997 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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