Question

13ft ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2(ft)/(s), how fast will the bottom of the ladder be moving away from the wall when the top is 5ft above the ground?

Answer 1

The bottom of the ladder will be moving away from the wall at a rate of 2 ft/s when the top is 5 ft above the ground.

Step-by-step explanation:

To find the speed at which the bottom of the ladder is moving away from the wall, we can use related rates. Let's use the Pythagorean theorem to relate the height of the ladder (y), the distance from the wall (x), and the length of the ladder (13 ft):

x^2 + y^2 = 13^2

Differentiating both sides of the equation with respect to time (t) gives:

2x(dx/dt) + 2y(dy/dt) = 0

We are given that dy/dt = -2 ft/s when y = 5 ft. Plugging these values into the equation, we can solve for dx/dt:

2(5)(dx/dt) + 2(5)(-2) = 0

(10)(dx/dt) - 20 = 0

10(dx/dt) = 20

dx/dt = 20/10

dx/dt = 2 ft/s

Therefore, the bottom of the ladder is moving away from the wall at a rate of 2 ft/s when the top is 5 ft above the ground.