A street light is at the top of a 13 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 feet from the base of the pole?
draw a diagram. Using similar triangles, you can see that if her shadow has length s, then when she is x feet from the pole,
s/6 = (x+s)/13
6(x+s) = 13s
6x+6s = 13s
s = 6/7 x
so at any time,
ds/dt = 6/7 dx/dt
so the shadow's length is increasing at 6/7 * 8 = 48/7 ft/s
now add that to the speed of the woman, and the tip of the shadow is moving at 104/7 ft/s
or,
let s be the distance of the shadow's tip from the pole. Then when the woman is x ft from the pole, we have
(s-x)/6 = s/13
13(s-x) = 6s
13s-13x = 6s
s = 13/7 x
when x=45, we have
ds/dt = 13/7 * 8 = 104/7 ft/s