My problem is that a stft above the sidewalk. a man 6ft tall walks away from the light at the rate of 3ft/sec. at what rate is his tip of shadow moving
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To find the rate at which the tip of the shadow is moving, we can use similar triangles and the concept of related rates. Here's how you can approach it:
Let's assume that the height of the streetlight is h and the length of the shadow is s. The man's height is 6ft.
Now, we can set up the following proportion:
h / s = 6 / (s + x)
Where x represents the distance the man has walked horizontally away from the light and will also represent the length of the shadow.
To find the related rates, take the derivative of both sides of the equation with respect to time (t):
(dh/dt) / (ds/dt) = 6 / (ds/dt + dx/dt)
We know that dx/dt (the rate at which the man is walking away from the light) is given as 3 ft/sec, so we can substitute it into our equation:
(dh/dt) / (ds/dt) = 6 / (ds/dt + 3)
Now, we need to figure out the value of (dh/dt), which represents the rate at which the top of the shadow is moving. Let's solve for it:
(dh/dt) = (6 / (ds/dt + 3)) * (ds/dt)
Now, we can plug in the given values to find the rate at which the top of the shadow is moving:
(dh/dt) = (6 / (ds/dt + 3)) * (ds/dt)
(dh/dt) = (6 / (6 + 3)) * 3
(dh/dt) = (6 / 9) * 3
(dh/dt) = 2 * 3
(dh/dt) = 6 ft/sec
Therefore, the rate at which the top of the shadow is moving is 6 ft/sec.