Question

A ladder 25 m long leans against a vertical all. If the lower end is being moved away from the wall at the rate 6m/sec, how fast is the height of the top decreasing when the lower end is 7m from the wall?

Answer 1

at the moment in question, the ladder top is 24m up the wall.

x^2+y^2 = 25^2
2x dx/dt + 2y dy/dt = 0
2(7)(6) + 2(24) dy/dt = 0
dy/dt = -4/7 m/s

Answer 2

start by drawing a diagram. you should notice that you have a triangle with the ladder as the hypoteneuse. you are given that the lower end is moving away from the wall at 6m/sec, so your dx/dt =6.
To find y when x=7:
x^2 +y^2 = 25^2
7^2 +y^2 = 625
49 + y^2 = 625
y^2 = 576
y = 24
Then, since we know that x is changing:
x^2 + y^2 = 25^2
2x(dx/dt) + 2y(dy/dt) = 0 divide by 2:
x(dx/dt) + y(dy/dt) = 0
plug in values that you know:
7(6) + 24(dy/dt) = 0
24(dy/dt) = -42
dy/dt = -1.75 m/sec