Calculus beggining question. (Explanation Needed) A real challenge!?

Question

A light is at the top of a 16- foot pole. A boy 5 ft tall walks away from the pole at a rate of 4 ft/sec. At what rate is the tip of his shadow moving when he is 18 ft from the pole? At what rate is the length of his shadow increasing?

Answer 1

Let's define some variables and constants:
vb = velocity of boy = 4 ft/sec
hb = height of boy = 5 ft
hl = height of lamp
lb = length of boy's shadow
ll = length from the lamp to the end of the boy's shadow
db = distance from lamp to boy
a = angle of light to ground

Since we want to find the length that the shadow is increasing, and the length that the tip of the shadow moves, first, let's define the length of the shadow:

tan a = opposite/adjacent
tan a = hb/lb = hl/ll
lb = hb * ll / hl
lb = 4 * ll / 16

Note: for the problem, ll = 18 ft + the length of the shadow, because the boy is 18 ft away from the pole.

lb = 4 * ll / 16
lb = 1/4 (18 + lb)
lb = 4.5 + 1/4 lb

The length of the shadow is increasing at the derivative of the length of the shadow: dlb/dt = 1/4 ft/sec (solution)

The tip of the shadow is increasing at the rate that the boy moves + the rate that the shadow lengthens = 4 + 1/4 = 4 1/4 ft /sec (solution)