how fast is the top of the ladder when the bottom of the ladder is 9 ft. from the wall?

Question

The ladder 15 ft. long rests against a wall. if the bottom of the ladder slides away from the wall at a rate of 1 ft./s. how fast is the top of the ladder when the bottom of the ladder is 9 ft. from the wall?

A.    0.75 ft/s

B.    0.25 ft/s

C.    0.50 ft/s

Answer 1

The ladder and the wall form a right triangle, with the hypotenuse equal to 15 feet.

Let y be the position of the top of the ladder against the wall and x be the distance the bottom of the ladder is sliding away from the wall. The problem is asking for dy/dx when x = 9 feet.

Using the Pythagorean forumula:

x²+y²=15² = 225

Taking the implicit derivative:

2x + 2y dy/dt = 0

dy/dt = -x/y

When x = 9, y = √(225-81) = √144 = ±12. We only need the positive value since y is a distance.

dy/dt = -9/12 =-3/4 ft/s = -0.75 ft/s, which is Answer A. This is negative, because the ladder is sliding down the wall.

Answer 2

I've always solved this by first writing y in terms of x.

Implicit differentiation is much easier

x^2 + y^2 = 15^2

2xdx + 2ydy = 0

dy/dx = -x/y

j = 12 (a 9-12-15 Pythagorean triple)

dy/dt = dy/dx dx/dt = -x/y dx/dt = -9/12 * 1 = -3/4

.75 ft/sec down the wall

A