A street light is at the top of a 20 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 50 feet from the base of the pole?
To find how fast the tip of the woman's shadow is moving, we need to use similar triangles and relate the movement of the woman to the movement of her shadow.
Let's define a few variables:
- h: height of the pole (20 ft)
- s: height of the woman (6 ft)
- x: distance between the woman and the pole base
- y: length of the woman's shadow cast on the ground
- z: distance between the tip of the shadow and the pole base
From similar triangles, we know that the ratio of the height of the pole to the length of its shadow is the same as the ratio of the height of the woman to the length of her shadow:
h / y = s / (y + z)
We want to find dz/dt, the rate at which the distance between the tip of the shadow and the pole base is changing when x = 50 ft.
First, let's differentiate both sides of the equation with respect to time (t):
d(h / y)/dt = d(s / (y + z))/dt
To simplify the equation, we need to eliminate y and express everything in terms of x:
From similar triangles, we have:
y / x = (h + s) / h
Rearranging this equation, we can express y in terms of x:
y = x * h / (h + s)
Now, we can substitute this expression for y in the rate of change equation:
d(h / (x * h / (h + s)))/dt = d(s / (x * h / (h + s) + z))/dt
Simplifying further, we have:
d(1 / (x / (h + s)))/dt = d(1 / ((x * h + (h + s)z) / ((h + s)(h + s))))/dt
Now, let's rearrange the equation to isolate dz/dt, the rate we're trying to find:
(d(1 / (x / (h + s)))/dt) * ((h + s)(h + s)) = d(1 / ((x * h + (h + s)z)))/dt
We have an expression for the rate of change of the tip of the shadow (dz/dt):
dz/dt = ((d(1 / (x / (h + s)))/dt) * ((h + s)(h + s))) / d(1 / ((x * h + (h + s)z)))/dt
Now, we need to substitute the given values into this equation to find the rate at which the tip of her shadow is moving when she is 50 feet from the base of the pole.
If the shadow has length s
and the woman is x feet from the pole, then
s/6 = (s+x)/20
s = 3/7 x
so, ds/dt = 3/7 dx/dt = 3
So, the tip of the shadow moves at 7+3 ft/s