HMMT 二月 2001 · CALC 赛 · 第 4 题
HMMT February 2001 — CALC Round — Problem 4
题目详情
英文原题
- What is the size of the largest rectangle that can be drawn inside of a 3-4-5 righttriangle with one of the rectangle’s sides along one of the legs of the triangle?
解析
英文解析
- What is the size of the largest rectangle that can be drawn inside of a 3-4-5 righttriangle 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 isy (4 / 3)(3 − y ) = 4 y − (4 / 3) y . The derivative of this is 0 when 4 − (8 / 3) y = 0, or y = 3 / 2,2
giving an area of 3.
Or, if you prefer, suppose the rectangle has a side of length x along the leg of length 4. Thenthe area is x (3 − (3 / 4) x ) = 3 x − (3 / 4) x . The derivative of this is 0 when 3 − (3 / 2) x = 0,2
or x = 2, again giving an area of 3 .