Question

A flood lamp is installed on the ground 200 feet from a vertical wall. A six-foot-tallman is walking towards the wall at the rate of 30 feet per second. How fast is the tip of his shadow moving down the wall when he is 50 feet from the wall

Answer 1

Answer:

The rate at which the tip of his shadow is moving down is $\frac{24}{15} ft/sec$

Step-by-step explanation:

Given - A flood lamp is installed on the ground 200 feet from a vertical wall. A six-foot-tall man is walking towards the wall at the rate of 30 feet per second.

To find - How fast is the tip of his shadow moving down the wall when he is 50 feet from the wall ?

Proof -

From the given information, the figure becomes

Triangle ABC and Triangle DBE are similar triangle

AC/DE = BC/BE

⇒h/6 = 200/(200-x)

⇒h = 1200/(200 -x)

Now,

Differentiate h with respect to t, we get

$\frac{dh}{dt} = 1200(-\frac{1}{(200-x)^{2} } )(-1)\frac{dx}{dt}$

⇒$\frac{dh}{dt} = (\frac{1200}{(200-x)^{2} } )\frac{dx}{dt}$

Now,

If the rate of the tip is moving down the wall

At x = 50 feet, dx/dt = 30 feet per second

So,

⇒$\frac{dh}{dt} = (\frac{1200}{(200-50)^{2} } )(30)$

⇒$\frac{dh}{dt} = \frac{1200}{(150)^{2} } (30)$

⇒$\frac{dh}{dt} = \frac{1200}{(150)(150) } (30)$

⇒$\frac{dh}{dt} = \frac{24}{15 }$

So, we get

The rate at which the tip of his shadow is moving down is $\frac{24}{15} ft/sec$