HMMT 二月 1998 · CALC 赛 · 第 7 题
HMMT February 1998 — CALC Round — Problem 7
题目详情
英文原题
Question Seven . [5 points]
A parabola is inscribed in equilateral triangle ABC of side length 1 inthe sense that AC and BC are tangent to the parabola at A and B ,
respectively:
Find the area between AB and the parabola.
解析
英文解析
- Problem: A parabola is inscribed in equilateral triangle ABC of side length 1 in the sense that ACand BC are tangent to the parabola at A and B , respectively. Find the area between AB and the parabola.
√
Solution: Suppose A = (0 , 0), B = (1 , 0), and C = ( , ). Then the parabola in question goes through 13
2 2
√ √
(0 , 0) and (1 , 0) and has tangents with slopes of 3 and − 3, respectively, at these points. Suppose thedyparabola has equation y = ax + bx + c . Then = 2 ax + b .2
√ √dxdy
At point (0 , 0), = b . Also the slope at (0 , 0), as we determined earlier, is 3. Hence b = 3. Similarly,
√ √dxdyat point (1 , 0), = 2 a + b . The slope at (1 , 0), as we determined earlier, is − 3. Then a = − 3.
√ √dx
Since the parabola goes through (0 , 0), c = 0. Hence the equation of the parabola is y = − 3 x + 3 x .2
The desired area is simply the area under the parabolic curve in the interval [0 , 1].
∫
1 ( )
√
√ √
2 3
Hence − 3 x + 3 x dx = .
06