A streetlight is mounted at the top of a 20-foot-tall pole. A person walks away from the pole at a speed of 4 feet per second. How fast is the tip of their shadow on the ground moving when they are 15 feet from the pole?
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We can solve this problem using similar triangles. Let's set up a right triangle with the streetlight, the pole, and the person's shadow. The height of the pole is 20 feet, and the distance from the pole to the person is changing at a rate of 4 feet per second. Let's call this distance x.
The length of the person's shadow on the ground is also changing as the person moves away from the pole. Let's call this length y.
Since the triangles formed by the person, the pole, and the shadow are similar, we can set up the following proportion:
y / 20 = x / (20 + x)
To find how fast the tip of the shadow is moving, we need to find dy/dt when x = 15.
Differentiating the equation above implicitly with respect to time, we get:
(20 + x)(dy/dt) = 20(dx/dt) + x(dx/dt)
When x = 15 feet, we can substitute this into the equation above to solve for dy/dt:
(20 + 15)(dy/dt) = 20 * 4 + 15 * 4
35(dy/dt) = 80 + 60
35(dy/dt) = 140
dy/dt = 140 / 35
dy/dt = 4
So, the tip of their shadow on the ground is moving at a rate of 4 feet per second.
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