# calculus

A street light is at the top of a 15 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving (away from the woman) when she is 50 feet from the base of the pole?
The tip of the shadow is moving at ____ ft/sec.

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1. draw a diagram. Let
z = distance from pole
Then using similar triangles,

x/6 = (z+x)/15
15x = 6z+6x
z = 3/2 x
so, dz/dt = 3/2 dx/dt
since dz/dt = 7, dx/dt = 14/3
so the tip of the shadow is moving at 7 + 14/3 = 35/3 ft/s

the distance from the pole does not matter for the speed.

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