A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 20 ft/sec. How fast must she let out the string when the kite is 600 ft away from her

Answer: [tex]10\sqrt3 ft/s[/tex] Step-by-step explanation:We are given that Height of kite from ground=300 ft In triangle ABC, AB=300 ft Let BC=h ft , AC=y=600 ft [tex]\frac{dh}{dt}=20 ft /s[/tex]We have to find the rate at which she must let out the string when the kite is 600 ft away from her.By pythagorous theorem [tex](300)^2+h^2=(600)^2[/tex][tex]90000+h^2=360000[/tex][tex]h^2=360000-90000=270000[/tex][tex]h=300\sqrt3 ft [/tex][tex](300)^2+h^2=y^2[/tex]Differentiate w.r.t t[tex]2h\frac{dh}{dt}=2y\frac{dy}{dt}[/tex][tex]h\frac{dh}{dt}=y\frac{dy}{dt}[/tex]Substitute the values then we get [tex]300\sqrt3\times 20=600\times \frac{dy}{dt}[/tex][tex]\frac{dy}{dt}=\frac{300\sqrt3\times 20}{600}=10\sqrt3 ft /s[/tex]Hence, she must let out the string at the rate [tex]10\sqrt3 ft/s[/tex] when the kite is 600 ft away from her.