Math Differentiation problem (20)

2008-10-24 3:16 am

A baseball diamond us a square with side 90ft. A batter hits the bakk and runs toward first base with a speed of 24 ft/s.

a) At what rate is his distance from second base decreasing when he is halfway to first base.

b) At what rate is his distance from third base increasing at the same moment?

use differentiation, thank you

回答 (1)

用戶10012

2008-10-24 6:11 am

✔ 最佳答案

Let distance of the batter from the hitting point be s. And let distance between the second base and the batter be y. Therefore, by Pythagoras theorem,
y^2 = 90^2 + (90 - s)^2
2y(dy/dt) = 2(90 -s)(-ds/dt).
Now ds/dt = 24, s = 45, y = sqrt(90^2 + 45^) = 45sqrt 5.
So dy/dt = (90- 45)(-24)/45sqrt 5 = -24/sqrt 5 that is decreasing at a rate of 24/sqrt5 ft/s.
Let distance between batter and third base = x. Again by Pythagoras theorem,
x^2 = 90^2 + s^2
2x(dx/dt) = 2s(ds/dt)
Now x = sqrt(90^2 + 45^2) = 45sqrt 5.
So dx/dt = 45(24)/45sqrt 5 = 24/sqrt 5 increasing.

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