Math
posted by Teri on .
a light shines from the top of a pole 50ft high. a ball is dropped from the same height from a point 30 ft away from the light. how fast is the ball's shadow moving along the ground 1/2 sec later? (assume the ball falls at a distance of s=16t^2 in t sec)

when the ball is at height h, the shadow is at distance x from the base of the light pole. Using similar triangles,
x/50 = (x30)/h
h = 50(x30)/x = 50  1500/x
now, h = 50s = 5016t^2, so
dh/dt = 32t, and
h(1/2) = 50  16(1/4) = 46
x/50 = (x30)/46, so x=375
32t = 1500/x^2 dx/dt
at t = 1/2,
16 = 1500/(140625) dx/dt
dx/dt = 1500
so, the shadow is moving toward the pole at 1500 ft/s