A street light is at the top of a 12 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 7 ft/sec along a straight path.

Question

A street light is at the top of a 12 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 when she is 35 feet from the base of the pole?

Answer 1

7 feet per second

is this a trick question?

she's moving at 7 feet per second, and the tip of her shadow moves at the same speed. 7 ft/sec

think of it all as a right triangle, with the pole as one side and the ground the other side.

if the hypotenuse formed is somehow considered the lengthof the shadow, then

h^2 = 12^2 + 70^2

h = sqr(144+4900) = sqr(5044) = 71.02 feet

h^2 = g^2 + p^2

g=distance from pole bottom to tip of shadow

p=12 feet, the pole size, the height of the right triangle

h=the hypotenuse of the right triangle, from top of the pole to the tip of the shadow

take the derivative with respect to time

2hh' = 2gg' = 2pp'

g'=14, p'=0

h=71.02, g = 70, p=12

71.02h' =70(14)

h' = 70(14)/71.02 = 13.8 feet/second

but only half of that could really somehow be considered a "shadow" so 13.8/2 = 6.9 ft/sec which is virtually the same as 7 ft/sec anyway

the angle formed by the ground and the hypotenuse has tangent theta = 12/70 = 6/35. theta = 9.73 degrees

theta' = -1/30 degree per second as the woman walks 7 ft/sec