Answer to Question #280918 in Calculus for Kelvin lee Boacon

Question #280918

2. 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?

1
Expert's answer
2021-12-20T13:35:59-0500


Pythagorean Theorem


L2=x2+1502L^2=x^2+150^2

Differentiate both sides with respect to tt


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

Solve for dLdt\dfrac{dL}{dt}


dLdt=(xL)dxdt\dfrac{dL}{dt}=(\dfrac{x}{L})\dfrac{dx}{dt}

Substitute L=x2+1502L=\sqrt{x^2+150^2}


dLdt=(xx2+1502)dxdt\dfrac{dL}{dt}=(\dfrac{x}{\sqrt{x^2+150^2}})\dfrac{dx}{dt}

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

If x=250 ftx=250\ ft


dLdt=(2502502+1502)(20)\dfrac{dL}{dt}=(\dfrac{250}{\sqrt{250^2+150^2}})(20)

dLdt=10034 ft/sec17.15 ft/s\dfrac{dL}{dt}=\dfrac{100}{\sqrt{34}}\ ft/sec\approx17.15\ ft/s



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