HMMT 二月 2001 · CALC 赛 · 第 4 题

HMMT February 2001 — CALC Round — Problem 4

What is the size of the largest rectangle that can be drawn inside of a 3-4-5 right triangle with one of the rectangle’s sides along one of the legs of the triangle?

解析

What is the size of the largest rectangle that can be drawn inside of a 3-4-5 right triangle with one of the rectangle’s sides along one of the legs of the triangle? Solution: Clearly one vertex of the rectangle will be at the right angle. Position the triangle with the leg of length 4 along the x -axis and the leg of length 3 along the y -axis. Then the hypotenuse is along the line y = 3 − (3 / 4) x . Suppose the rectangle has a side of length y along the leg of length 3. Then the area is 2 y (4 / 3)(3 − y ) = 4 y − (4 / 3) y . The derivative of this is 0 when 4 − (8 / 3) y = 0, or y = 3 / 2, giving an area of 3. Or, if you prefer, suppose the rectangle has a side of length x along the leg of length 4. Then 2 the area is x (3 − (3 / 4) x ) = 3 x − (3 / 4) x . The derivative of this is 0 when 3 − (3 / 2) x = 0, or x = 2, again giving an area of 3 .