Answer to Question #281953 in Calculus for Bret

Question

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

Answer 1

Pythagorean Theorem

$$L^2=x^2+150^2$$

Differentiate both sides with respect to $t$

$$2L(\dfrac{dL}{dt})=2x(\dfrac{dx}{dt})$$

Solve for $\dfrac{dL}{dt}$

$$\dfrac{dL}{dt}=(\dfrac{x}{L})\dfrac{dx}{dt}$$

Substitute $L=\sqrt{x^2+150^2}$

$$\dfrac{dL}{dt}=(\dfrac{x}{\sqrt{x^2+150^2}})\dfrac{dx}{dt}$$

Given $\dfrac{dx}{dt}=20 ft/sec$

If $x=250\ ft$

$$\dfrac{dL}{dt}=(\dfrac{250}{\sqrt{250^2+150^2}})(20)$$ $$\dfrac{dL}{dt}=\dfrac{100}{\sqrt{34}}\ ft/sec\approx17.15\ ft/s$$