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A street light is at the top of a 15 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 4 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 ft from the base of the pole?
First we need to find θ. From the picture we can see that:
Now we need to relate the woman's velocity to the change in θ. The distance from the base of the pole to the woman is:
However, we're not actually interested in the length of the shadow, but rather the distance of the tip of the shadow from the base of the street light. Call this distance x:
Then the first derivative of x with respect to t will give us the change in the distance of the tip of the shadow from the base of the street light:
So when the woman is 45 ft from the base of the pole, the tip of her shadow is moving 10/3 = 3.3 ft/sec. Note that this is the same as dw/dt.
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