i need help in this calculus question

Question

A street light is at the top of a 14 ft tall pole. A woman 6 ft tall 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 30 ft from the base of the pole?

Answer 1

Let the base of the pole be at 0,0). The light is at (0, 14). The woman is standing at (x, 0); the top of her head is at (x, 6). You can draw a line from (0, 14) to (x, 6) and see where it meets the x axis. We will let this point be (z, 0)

The line through (0, 14) and (x, 6) has slope (6 - 14)/(x-0) = -8/x

As it goes through (0, 14), it has x-coefficient 14, so the equation is y = (-8/x)x + 14

Then, solving for y = 0 (the tip of the shadow), 0 = (-8/x)z + 14

14 = (8/x)z

z = 14/(8/x)

z = 14/8 x = 7/4 x

It is pretty easy to check that this is correct by drawing a picture and seeing that we have similar triangles when z has this coordinate. This might actually lead you to create this equation for z by showing that these triangles are similar.

dx/dt = 8 ft/sec

If z = 7/4x , dz/dx = 7/4

Then, dz/dt = dz/dx dx/dt = 7/4 * 8 ft/sec = 14 ft/sec