Related Rates Problem – Lantern?

Question

A 60 foot rope is looped thru a pulley 36 feet above the ground. A lantern is attached to one end. The other end is held by a man six feet tall. The man starts walking away from the point on the ground directly beneath the pulley (and beneath the lantern) at a rate of five feet per second. He is holding onto the rope head-high. When the man is 40 feet away from the point on the ground directly beneath the pulley, what is the rate of change of the length of his shadow that is cast by the lantern?

Best answer is correct and clearly presented and explained.
Have fun.

Answer 1

I'll bite.

The pulley separates the rope into two parts: 60-L and L such that 36-L is the height of the Lantern.

Let x be the horizontal distance between the man and the point on the ground directly beneath the pulley, and s be the length of his shadow.

By Pythagorean theorem,
(60-L)^2 = x^2+(36-6)^2......(1)

Differentiate (1) with respect to time and solve for L',
L' = -4 ft/sec......(2)

By triangle similarity,
s/6 = x/(30-L)......(3)

Differentiate (3) with respect to time, plug in (2), and solve for s',
s' = -0.9 ft/sec......(4)

The negative sign in (4) means the length of his shadow decreases with time at the point.

Conclusion: The rate of change of the length of his shadow that is cast by the lantern is s' = -0.9 ft/sec.