Question

A 5.5 foot tall woman walks at 6ft/s torward a street light that is 16.5 ft above the ground. what is the rate of change of the length of her shadow when she is 14ft from the street light? At what rate is the tip of her shadow moving?

How do I get the equation for this and how do I work it out?

Answer 1

two similar triangles

First the stationary situation
x/5.5 = (x+14)/16.5

5.5(x+14) = 16.5 x

11 x = 77
x = 7

now start moving
x = length of shadow
y = woman to pole

(x+y)/16.5 = x/5.5
16.5 x = 5.5 x + 5.5 y
11 x = 5.5 y
2 x = y
so
2 dx/dt = dy /dt
if dy/dt = -6 then dx/dt = -3 ft/s

then for tip
d tip/dt = dy/dt + dx/dt
= -3-6 = -9 ft/s