a building is 36 ft tall. A pulley is attached to the top of the building. A rope is looped through the pulley. One end of the rope is attached to the lantern that hangs vertically parallel to the side of the building. The rope passes up vertically from the lantern, through the pulley, and then diagonally downwards. The other end of the rope is being held by a man who is 6 ft tall. The man holds the rope on top of his head and walks away from the building, causing the lantern to rise. The rope is 60 ft. At a certain instant the man is walking away at a rate of 5 ft per second and the man is 40 ft away. Find the rate change of the length of the mans shadow.

Question

a building is 36 ft tall. A pulley is attached to the top of the building. A rope is looped through the pulley. One end of the rope is attached to the lantern that hangs vertically parallel to the side of the building. The rope passes up vertically from the lantern, through the pulley, and then diagonally downwards. The other end of the rope is being held by a man who is 6 ft tall. The man holds the rope on top of his head and walks away from the building, causing the lantern to rise. The rope is 60 ft. At a certain instant the man is walking away at a rate of 5 ft per second and the man is 40 ft away. Find the rate change of the length of the mans shadow.

Answer 1

LOVE IT!

Assuming the lantern is in fact shining, we have

If the length of rope from the man to the pulley is z, and the man is x ft away from the wall, we have

x^2+30^2 = z^2
z dz/dt = x dx/dt
when x=40, z=50 and the lantern is at height 36-10 = 26

If the lantern is at height h, and the man's shadow is of length s, then using similar triangles, we have

s/6 = (s+x)/h
= (s+x)/(36 - (60-z)) = (s+x)/(z-24)
= (s+x)/(√(x^2+900)-24)
s(1/6 - 1/(√(x^2+900)-24)) = 6x/(√(x^2+900)-30)
ds/dt = -180/(x^2+900-30√(x^2+900)) dx/dt
= -180/(z^2-30z) dx/dt

So, plug in the numbers and we have

ds/dt = -180/(50^2-30*50 = -180/1000 = -0.18 ft/s

whew - better check my math

As a check, using z instead of all the radicals, we have

s/6 = (s+x)/(z-24)

I guess you can verify that ds/dt comes out the same. Just remember that both s and z are functions of x.

Answer 2

To find the rate of change of the length of the man's shadow, we need to consider similar triangles formed by the building, the man, and his shadow.

Let's label the following:
- h: height of the building = 36 ft
- x: horizontal distance between the man and the building (length of the shadow)
- y: vertical distance between the base of the building and the man's shadow

Now, we can set up a proportion using the similar triangles:
h / x = (h + y) / (x + 40)

We know that h = 36 ft and the man is 6 ft tall, so y = h - 6 = 36 - 6 = 30 ft.

Substituting the values, we have:
36 / x = (36 + 30) / (x + 40)

Now, we can solve for x by cross-multiplying:
36(x + 40) = 66x

36x + 1440 = 66x

30x = 1440

x = 1440 / 30

x ≈ 48 ft

Now, we can find the rate of change of the length of the man's shadow (dx/dt) using the given information that the man is walking away from the building at a rate of 5 ft/s:
dx/dt = 5 ft/s

Therefore, the rate of change of the length of the man's shadow is 5 ft/s.