Question

Upper A 16​-foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 2 ​feet/second, how fast is the top of the ladder moving down when the foot of the ladder is 5 feet from the​ wall?

Answer 1

Answer:

The ladder is moving at the rate of 0.65 ft/s

Explanation:

A 16​-foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 2 ​feet/second. We need to find the rate at which the top of the ladder moving down when the foot of the ladder is 5 feet from the​ wall.

The attached figure shows whole description such that,

$x^2+y^2=256$.........(1)

$\dfrac{dx}{dt}=2\ ft/s$

We need to find, $\dfrac{dy}{dt}$ at x = 5 ft

Differentiating equation (1) wrt t as :

$2x.\dfrac{dx}{dt}+2y.\dfrac{dy}{dt}=0$

$2x+y\dfrac{dy}{dt}=0$

$\dfrac{dy}{dt}=-\dfrac{2x}{y}$

Since, $y=\sqrt{256-x^2}$

$\dfrac{dy}{dt}=-\dfrac{2x}{\sqrt{256-x^2}}$

At x = 5 ft,

$\dfrac{dy}{dt}=-\dfrac{2\times 5}{\sqrt{256-5^2}}$

$\dfrac{dy}{dt}=0.65$

So, the ladder is moving down at the rate of 0.65 ft/s. Hence, this is the required solution.