Question

A point source of light is hung $30$ feet directly above a straight horizontal path on which a man of 6 feet in height is walking . how fast is the man 's shadow lengthening and how fast the tip of the shadow is moving when he is walking away from the light at the rate of $100ft/min$.

Answer 1

Let $s$ be the point pof source of light

Rate of walking $=100ft/min$

In $△ASC$ & $△BDC$

$\frac{AS}{BD}=\frac{AC}{BC}$

$\frac{30}{6}=\frac{x+y}{y}$

$\Rightarrow 5y=x+y$

$\Rightarrow 4y=x$

$4\frac{dy}{dx}=\frac{dx}{dt}=100\Rightarrow \frac{dy}{dt}=25ft/min$ (Length of shadow moves at rate $dy/dt$)

Now Let $AC=z$

Now $BC=AC-AB=z-x$

Now Again IN $△ASC$ & $△BDC$

$\frac{AS}{BD}=\frac{AC}{BC},\frac{30}{6}=\frac{z}{z-x}\Rightarrow 4z=5x$

Now $\frac{dz}{dt}=125ft/min$ (Rate of change of shadow)