Answer to Question #174357 in Calculus for Holden Giles Cabrito

Question

A boy flying a kite pays out string at a rate of 2 ft/sec as the kite moves 

horizontally at an altitude of 100 ft. Assuming there is no sag in the 

string, find the rate at which the kite is moving when 125 ft of string 

have been paid out.

Answer 1

A right angle triangle is formed in the air. The string forms the hypotenuse, the horizontal distance it is moving forms the base and the altitude forms the height. The kite stays at a constant height of 100ft. The string is 125 feet and moving out at 2 ft/sec.

let $y=100ft$ , $z=125ft$ and ${dz \over dt}=2ft/sec$ , $x=$ Horizontal distance moved by the kite and ${dx \over dt}=$ The rate at which the kite is moving

$\therefore x^2+y^2=z^2 ; x=\sqrt{125^2 - 100^2}=75ft$

differentiating the above equation with respect to t we have

$2x{dx \over dt} + 0=2z{dz \over dt}$

${dx \over dt}={z \times {dx \over dt} \over x}={125 \times 2 \over 75}=3.33ft/sec$

when 125ft of the string has been paid out, the kite moves at a speed of 3.33ft/sec