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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Let's assume that the person is standing at point P on the ground, which is 15 feet from the pole. Let Q be the tip of the person's shadow on the ground, and let R be the top of the pole.
Using similar triangles, we can form the following ratio:
(PQ / QR) = (PR / QR)
We know that PQ is the length of the person's shadow, QR is the height of the pole (20 feet), and PR is the distance from the person to the top of the pole (20 - 15 = 5 feet).
Let's differentiate both sides of the equation with respect to time:
(d(PQ) / dt) / QR = (d(PR) / dt) / QR
The derivative of PQ with respect to time represents the rate at which the shadow's length is changing, which is what we want to find.
(d(PQ) / dt) = (d(PR) / dt) * (QR / QR)
Since QR = 20, and we know that (d(PR) / dt) = 4 ft/s (the speed at which the person is walking away from the pole), we can substitute those values into the equation above:
(d(PQ) / dt) = 4 ft/s * (20 ft / 20 ft)
(d(PQ) / dt) = 4 ft/s
Therefore, the tip of the person's shadow on the ground is moving at a constant speed of 4 ft/s.
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