A lamp post is 6m tall. Frank is 1.6 m and walks toward the post at .5m/s. How fast is the tip of his shadow moving when he is 8 m from the pole?
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To find the speed at which the tip of Frank's shadow is moving when he is 8m from the pole, we can use similar triangles.
Let's define some variables:
- The height of the lamp post is h = 6m.
- The height of Frank is f = 1.6m.
- The distance of Frank from the lamp post is x = 8m.
- The distance of the tip of Frank's shadow from the lamp post is y (which is what we want to find).
First, we can set up a proportion based on the similar triangles formed by the lamp post, Frank, and his shadow:
(f / h) = (y / (x + y))
Now, we can solve this equation to find y:
(f / h) = (y / (x + y))
(y / (x + y)) = f / h
y = (f / h) * (x + y)
Now, let's substitute the given values into the equation:
y = (1.6 / 6) * (8 + y)
Next, we can solve for y by multiplying both sides of the equation by (8 + y):
y = (1.6 / 6) * (8 + y)
6y = 1.6(8 + y)
6y = 12.8 + 1.6y
Simplifying the equation:
6y - 1.6y = 12.8
4.4y = 12.8
y = 12.8 / 4.4
y ≈ 2.91m
Now that we have found the value of y, we can differentiate the equation with respect to time to find the rate at which y changes with respect to x.
Differentiating both sides of the equation:
d/dt(y) = d/dt((1.6 / 6) * (8 + y))
Now, let's differentiate the equation:
dy/dt = (1.6 / 6) * dy/dt(8 + y)
dy/dt = (1.6 / 6) * (0 + dy/dt(y))
dy/dt = (1.6 / 6) * (dy/dt(y))
Since we know that Frank is walking toward the post at a speed of 0.5m/s, dy/dt can be substituted with -0.5 (negative sign indicates a decreasing value):
-0.5 = (1.6 / 6) * dy/dt(y)
Now, we can solve for dy/dt(y) by multiplying both sides of the equation by (6 / 1.6):
dy/dt(y) = -0.5 * (6 / 1.6)
dy/dt(y) ≈ -1.875 m/s
So, the tip of Frank's shadow is moving at approximately -1.875 m/s when he is 8m from the pole. The negative sign indicates that the shadow is shrinking as Frank moves closer to the pole.