Final answer:
Using related rates, we can find that the top of the ladder is falling at a rate of -400/13 ft/s when the base is 50 ft from the building.
Step-by-step explanation:
To solve this problem, we can use related rates. Let's define some variables first: the height of the building (h) is 130 ft, the distance of the base of the ladder from the building (x) is 50 ft, and the distance the base is sliding away from the building (dx/dt) is 4 ft/s.
We are asked to find the rate at which the top of the ladder (h) is falling (dh/dt) when the base is 50 ft from the building.
Using the Pythagorean theorem, we can relate the height (h), the distance of the base (x), and the length of the ladder (L): L^2 = h^2 + x^2.
Taking the derivative of both sides with respect to time, we get 0 = 2h * dh/dt + 2x * dx/dt.
Plugging in the given values, we have 0 = 2(130) * dh/dt + 2(50) * 4.
Solving for dh/dt, we get dh/dt = -400/13 ft/s.
Therefore, the top of the ladder is falling at a rate of -400/13 ft/s when the base is 50 ft from the building.