Question

a boy flying a kite at a height of 150 ft. of the kite moves horizontally away from the boy at 20 ft/sec, how fast is the string being paid out when the kite is 250 ft from him?

Answer 1

As the kite moves horizontally at rate 20 ft/second, the string will be paid out at rate 16 ft/sec.

The boy, kite, and the ground point beneath the kite form a right triangle.

Let:

x = distance between the boy and the ground point beneath the kite

y = height of the kite

s = distance between the boy and the kite

Using the Pythagorean Theorem:

x² + y² = s²

x² + 150² = 250²

x² = 250² - 150²

x² = 200²

x = 200 ft.

Take the derivative with respect to t:

2x . dx/dt + 2y . dy/dt = 2s . ds/dt

From the problem, we know that: dx/dt = 20 ft/sec and dy/dt = 0 since the kite moves horizontally. Hence,

2.(200) . (20) + 2.(150) . 0 = 2. (250) . ds/dt

ds/dt = 200 . 20 / 250 = 16 ft/sec.

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