Hello, ginax7!

A tightrope is stretched 30 feet above the ground between two buildings which are 50 feet apart.

A tightrope walker, walking at a constant rate or 2 ft/ sec from point A to point B,

is illuminated by a spotlight 70 feet above point A.

How fast is the shadow of the tightrope walker's feet moving along the ground

when she is midway between the buildings?

I am so confused. Where do I start? . . . . Make a sketch! Code:

L *
| *
| *
70 | *
| *
| * T
A * - - - - - o - - - - - - * B
| x * |
30 | * | 30
| * |
* - - - - - - - - - o - - *
C - - - - s - - - - E D
: - - - - - -50 - - - - - :

The tightrope walker is at $\displaystyle T$, having walked $\displaystyle x$ feet from $\displaystyle A$ to $\displaystyle T.$

The spotlight is at $\displaystyle L$, casting the shadow at $\displaystyle E$ on the ground.

Let $\displaystyle s \,= \,CE.$

From similar right triangles, we have: .$\displaystyle \frac{s}{100} \:=\:\frac{x}{70}\quad\Rightarrow\quad s \:=\:\frac{10}{7}x$

Differentiate with respect to time: .$\displaystyle \frac{ds}{dt} \:=\:\frac{10}{7}\,\frac{dx}{dt}$

Since $\displaystyle \frac{dx}{dt} \,= \,2$ ft/sec: . $\displaystyle \frac{ds}{dt} \:=\:\frac{10}{7}(2) \:=\:\frac{20}{7}$ ft/sec.