2007 iTest Problems/Problem 32
Problem
When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio to . How many integer values of k are there such that and the area between the parabola and the -axis is an integer?
Solution
The vertex of the quadratic is . Factoring the quadratic results in , so the two zeroes are and . Thus, the area of the rectangle is , so the area of the parabola is .
In order for to be an integer, must be a perfect square and a multiple of . Note that , so the values that work are . In total, there are values of that satisfy the conditions.
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 31 |
Followed by: Problem 33 | |
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