Question

a 13-foot ladder is leaning against a wall. if we pull the ladder away from the wall at a rate of 6ft/s, how fast is the top of the ladder moving down the wall when the bottom of the ladder is 12ft from the wall?

Answer 1

Below in bold.

Explanation:

The relation between the height of the ladder and distance from the ground is given by Pythagoras Theorem:

h^2 = 13^2 - L^2

h = (169 - L^2)^1/2

Finding the derivative:

dh/dL = 1/2(169 - L^2)^-1/2 * -2L

= -L / (169 -L^2)^1/2

dL/dt = 6

So, dh/dt = dh/dL * dL/dt

= -6 * L/ (169 - L^2)^1/2

= -6 * 12 / (169 - 12^2)^1/2

= -72/5 ft /second

= -14.4 ft/second.

It's negative because the distance from the ground is decreasing.