Answer
(a) The light is at a height of 15 feet.
The man is 6 feet tall.
The man is walking away from the pole at a rate of $5~ft/s$
(b) The unknown is $\frac{dz}{dt}$ where $z$ is the distance between the tip of the shadow and the pole.
(c) We can see a sketch below.
(d) $\frac{x}{6} = \frac{x+y}{15}$
$z = x+y$
(e) The tip of the shadow is moving at a speed of $~~\frac{25}{3}~ft/s$
Work Step by Step
(a) The light is at a height of 15 feet.
The man is 6 feet tall.
The man is walking away from the pole at a rate of $5~ft/s$
(b) The unknown is $\frac{dz}{dt}$ where $z$ is the distance between the tip of the shadow and the pole.
(c) We can see a sketch below.
(d) Using similar triangles:
$\frac{x}{6} = \frac{x+y}{15}$
$z = x+y$
(e) We can find $\frac{dx}{dt}$:
$\frac{x}{6} = \frac{x+y}{15}$
$15x = 6x+6y$
$9x = 6y$
$x = \frac{2}{3}~y$
$\frac{dx}{dt} = \frac{2}{3}~\frac{dy}{dt}$
$\frac{dx}{dt} = \frac{2}{3}~(5~ft/s)$
$\frac{dx}{dt} = \frac{10}{3}~ft/s$
We can find $\frac{dz}{dt}$:
$z = x+y$
$\frac{dz}{dt} = \frac{dx}{dt}+\frac{dy}{dt}$
$\frac{dz}{dt} = (\frac{10}{3}~ft/s)+(5~ft/s)$
$\frac{dz}{dt} = \frac{25}{3}~ft/s$
The tip of the shadow is moving at a speed of $~~\frac{25}{3}~ft/s$
