Question

A 5.5-foot-tall woman walks at 8 ft/s toward a street light that is 22 ft above the ground. What is the rate of change of the length of her shadow when she is 20 ft from the street light? At what rate is the tip of her shadow moving?

Answer 1

Answer: $\frac{8}{3}\ ft/s$

Step-by-step explanation:

Given

length of shadow =22 ft

height of woman =5.5 ft

speed of woman $\frac{dy}{dt}=8\ ft/s$

from the figure, we can write

$\frac{22}{x+y}=\frac{5.5}{x}$

at given instant

y=20 ft

so, using similar triangle properties

$\frac{22}{x+20}=\frac{5.5}{x}\\22x=5.5x+110\\16.5x=110\\x=6.67\ ft$

Again we can write

$\dfrac{22}{x+y}=\dfrac{5.5}{x}\\22x=5.5x+5.5y\\16.5x=5.5y\\3x=y\\\text{differentiate w.r.t time} \\3\frac{dx}{dt}=\frac{dy}{dt}\\\frac{dx}{dt}=\frac{1}{3}\times 8=\frac{8}{3}\ ft/s\ \text{away from the woman}$