Question

A ladder 25ft long is leaning against the wall of a house. The ladder is pulled away from the wall at a rate of 2 ft per second. How fast is the top of the ladder moving down the wall when its base is 7ft from the wall?

Answer 1

This is the classic problem that almost every text in Calculus uses to introduce "rate of change"

Make a diagram
Let the ladder by y ft above the ground, and
x ft away from the wall

so you have a right angled triangle
x^2 + y^2 = 25
2x dx/dt + 2y dy/dt = 0 (#1)

given : dx/dt = 2 ft/s
find : dy/dt when x = 7

when x = 7
49 + y^2 = 625
y = √576 = 24

back in #1
2(7)(2) + 2(24)dy/dt = 0
dy/dt = -28/48 = -7/12 ft/s

the negative indicates that the value of y is decreasing, or in other words, the ladder is dropping along the wall