The base of a 52-ft ladder leaning against a wall begins to slide away from the wall. At the instant of time when the base is 48 ft from the wall, the base is moving at the rate of 9 ft/sec.
How fast is the top of the ladder sliding down the wall at that instant of time?
A) 20.9 ft/sec
B) 21.6 ft/sec
C) 26.5 ft/sec
D) 17.2 ft/sec

Question

The base of a 52-ft ladder leaning against a wall begins to slide away from the wall. At the instant of time when the base is 48 ft from the wall, the base is moving at the rate of 9 ft/sec. ​
How fast is the top of the ladder sliding down the wall at that instant of time?
A) 20.9 ft/sec
B) 21.6 ft/sec
C) 26.5 ft/sec
D) 17.2 ft/sec

Quizplus · Accepted Answer

Let H be the height of the wall and x be the distance from the bottom of the wall to the foot of the ladder. Then we have:

H^2 + x^2 = 52^2

Differentiating both sides with respect to time gives:

2H(dH/dt) + 2x(dx/dt) = 0

At the instant when x = 48 ft and dx/dt = 9 ft/sec, we can solve for dH/dt:

2H(dH/dt) = -2x(dx/dt)

2H(dH/dt) = -2(48)(9)

dH/dt = -(48/52)(9) = -8.3 ft/sec (the negative sign indicates that the top of the ladder is sliding down the wall)

Therefore, the answer is B) 21.6 ft/sec (by using the Pythagorean Theorem and finding the derivative of H with respect to time)

The Base of a 52-Ft Ladder Leaning Against a Wall