A streetlight is mounted on a 15-foot pole. A person walks away from the pole at a rate of 5 feet per second. How fast is the tip of their shadow moving when they are 20 feet from the pole?
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Let's set up a right triangle to represent the situation. The height of the pole is 15 feet and the distance the person has walked away from the pole is 20 feet. Let's call the distance from the person to the tip of the shadow x.
Using the properties of similar triangles, the height of the person's shadow is x/20 * 15.
Now, let's differentiate both sides of the equation with respect to time to find the rate of change of the shadow's height.
d/dt(x/20 * 15) = d/dt(x) * 15/20
The rate at which the person is walking away from the pole is given as 5 feet per second. So, dx/dt = 5.
Plugging in this value, we get:
d/dt(x/20 * 15) = 5 * 15/20 = 3.75
Therefore, the tip of the person's shadow is moving at a rate of 3.75 feet per second.
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