Answer to Question #198453 in Calculus for Ann

Question #198453

A light on the ground at Lilys building is 30meters away from the building. Jade is 2 meters tall. She walks from the light to the building at 1 meters per second. How fast is the shadow of Jade on the building changing when she is 15 meters from the building.

Expert's answer

"\\text{By using similar triangle property,}\\newline\n\\frac{2}{x}=\\frac{h}{30}\\newline\nh=\\frac{60}{x}\\newline\n\\text{Differentiate wrt x,}\\\\\n\\frac{dh}{dx}=\\frac{-60}{x^2}\\\\\nGiven, \\frac{dx}{dt}=1m\/sec.\\newline\n\\text{\nwe need to find} \\frac{dh}{dt}.\\newline\nTherefore, \\frac{dh}{dt}=\\frac{dh}{dx}\u00d7\n\\frac{dx}{dt}\\newline\n=\\frac{-60}{x^2}\u00d71\\newline\n=\\frac{-60}{x^2}\\newline\n\\text{\nGiven, when she is 15 m from the building.} \\newline\n\\implies x=30-15=15m\\newline\nTherefore,\\newline\n\\frac{dh}{dt}=\\frac{-60}{15^2}=\\frac{-4}{15}=-0.267m\/sec"