An 18 foot ladder is leaning against a building if the bottom of the ladder is sliding along the pavement directly away from the building at 3 ft./s how fast is the top of the ladder moving down when the foot of the ladder is 2 feet from the wall

Question

An 18 foot ladder is leaning against a building if the bottom of the ladder is sliding along the pavement directly away from the building at 3 ft./s how fast is the top of the ladder moving down when the foot of the ladder is 2 feet from the wall

Answer 1

Let the distance between the top of the ladder and the wall be x ft.

From the Pythagorean theorem, we have:

x^2 + (18)^2 = (18 + 3t)^2

Differentiating both sides with respect to t, we get:

2x(dx/dt) = 2(18 + 3t)(3)

dx/dt = 9(18 + 3t)/x

Plugging in x = sqrt(18^2 - 2^2) = 3√10, and t = 2 into the equation, we get:

dx/dt = 9(18 + 3*2)/(3√10) = 90/√10 = 28.50 ft/s

Therefore, the top of the ladder is moving down at a rate of 28.5 ft/s.