Question

A 30 foot ladder is leaning against a house(see figure). if the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a a rate of r= 2x/(sqrt(900-x^2) ft/sec, where s ist the distance between the base of the ladder and the house. Find the rate r when x is 18 feet. Help!

Answer 1

what's the problem? Just plug in your value of x:

r = 2(18)/24 = 3/2

This may be clearer if we consider the height h up the wall.

x^2 + h^2 = 900
2x dx/dt + 2h dh/dt = 0
when x=18, h=24, and since dx/dt = 2,

18(2) + 24 dh/dt = 0
dh/dt = -18(2)/24 = -3/2

Note that we have a negative value for dh/dt, which indicates the height is decreasing as the base is pulled away. The formula they gave gives a positive value, but it is stipulated that the top is moving down at that rate.

Answer 2

I was confused about if I should just plug in the value or take the derivative of the function? But Thank you!