A street light is at the top of a 10.5 ft. tall pole. A man 5.4 ft tall walks away from the pole with a speed of 3.5 feet/sec along a straight path. How fast is the tip of his shadow moving when he is 47 feet from the pole?
Nevermind, I got it. it's 7.21
To find the rate at which the tip of the man's shadow is moving, we can use similar triangles and relate the rate at which the man is moving to the rate at which the shadow tip is moving.
Let's denote:
- h = height of the pole, which is 10.5 ft
- s = length of the shadow when the man is 5.4 ft away from the pole
- x = distance between the man and the pole at any given point in time
- y = length of the shadow when the man is x ft away from the pole
We are given that the man is walking away from the pole with a speed (rate of change) of 3.5 ft/sec. This means the rate at which x is changing with respect to time (dx/dt) is 3.5 ft/sec.
Now, let's set up a proportion using similar triangles:
(Height of the pole) / (Length of the shadow) = (Height of the man) / (Length of the shadow tip)
h / s = (man's height) / y
Since we know h = 10.5 ft and the man's height = 5.4 ft, we can rewrite the equation as:
10.5 / s = 5.4 / y
To find the length of the shadow when the man is 47 ft away (y), we can use the proportion above:
10.5 / s = 5.4 / y
Cross-multiplying:
10.5y = 5.4s
y = (5.4s) / 10.5
Now, differentiate both sides of the equation with respect to time:
d/dt (y) = d/dt [(5.4s) / 10.5]
dy/dt = (5.4/10.5) * ds/dt
We are given that ds/dt (rate of change of the length of the shadow) is the same as dx/dt (rate at which the man is moving), which is 3.5 ft/sec.
So, substituting the values:
dy/dt = (5.4/10.5) * 3.5
Calculating the value:
dy/dt = 1.8 * 3.5
dy/dt = 6.3 ft/sec
Therefore, the tip of the man's shadow is moving at a rate of 6.3 ft/sec when he is 47 ft away from the pole.