A 15-foot ladder is leaning against a house. If the tip of the ladder begins to slide down the house at a rate of 5 feet per second. What is the rate of change of the base of the ladder.

Question

A 15-foot ladder is leaning against a house. If the tip of the ladder begins to slide down the house at a rate of 5 feet per second, what is the rate of change of the base of the ladder when the base is 9 feet from the house?

Answer 1

Will need to start with Pythagorean theorem to relate the 3 sides of the triangle formed

Call y the height and x the distance from the base of the house to the base of the ladder and call c the length of the ladder

So x^2 +y^2 =15^2= 225. take the derivative of both sides with respect to time(t)

2x dx/dt + 2y dy/dt = 0 ( derivative of a constant is zero) so

x dx/dt + y dy/dt = 0 and we want to solve for dx/dy

The length of the ladder is constant at 15 so when x= 9 by Pythagorean theorem y=12

Do we have x and y and are given dy/dt = -5 ( as the ladder slides down the wall y is decreasing)

9 dx/dt + (-5)(12)= 0

9dx/dt = 60

dx/dt= 60/9 =20/3 feet/second