Final answer:
To determine the rate at which string must be let out for a kite flying at a constant height, we use the Pythagorean theorem and implicit differentiation. When 500 ft of string is out, and the horizontal distance increases at 25 ft/sec, the string must be let out at a rate of 41.25 ft/sec.
Step-by-step explanation:
Calculating the Rate at Which String is Let Out for a Kite
To determine how fast the girl must let out the string (rate of change of the string's length), we can apply the Pythagorean theorem to the right-angled triangle formed by the height of the kite, the horizontal distance, and the length of the string. If the kite flies at a constant height of 300 ft and is being carried away horizontally at 25 ft/sec, we can find the rate at which the string is let out when there is 500 ft of string already out.
Let h be the height of the kite, x be the horizontal distance, and s be the length of the string. Since the height is constant (300 ft), dh/dt is 0. The horizontal distance changes at 25 ft/sec (dx/dt = 25 ft/sec). We are looking for ds/dt when s = 500 ft.
By the Pythagorean theorem, s² = h² + x². Differentiating both sides with respect to time t, we get 2s ds/dt = 2x dx/dt. Since h is constant, its derivative is zero. Now we plug in the known values: s = 500 ft, h = 300 ft, and dx/dt = 25 ft/sec. Solving for ds/dt gives us the rate at which the string must be let out.
Thus, we find that when s = 500 ft, ds/dt (the rate at which the string must be let out) is 41.25 ft/sec.