# Few pratice problems that I'm having trouble with.

• Mar 25th 2007, 02:21 PM
mathman66
Few pratice problems that I'm having trouble with.
A lighthouse is located 3km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1km from P?

I don't get how to maximize in this equation.
A circle of radius r, you carve out a slice from the middle that makes angle 0<X<2pi, and has perimeter P. Maximize the area of the slice.
• Mar 25th 2007, 03:31 PM
Soroban
Hello, mathman66!

Quote:

A lighthouse is located 3km away from the nearest point P on a straight shoreline
and its light makes four revolutions per minute.
How fast is the beam of light moving along the shoreline when it is 1km from P?

Code:

```      L       *       |  *       | θ  *     3 |        *       |          *       |              *       * - - - - - - - - *       P        x        B```

The lighthouse is at L.
Its beam shine at point B on the shore, x km from point P.
Let θ = angle BLP.

The light makes 4 revolutions per minute.

We have: .tanθ .= .x/3 . . x .= .3·tanθ

Differentiate with respect to time: .dx/dt .= .3·secēθ·(dθ/dt) .[1]

When x = 1, we have a right triangle with angle θ.
The opposite side is 1, the adjacent side is 3.
. . Then the hypotenuse is √10.
Hence: .secθ = √10/3 .and .secēθ = 10/9

Substitute into [1]: .dx/dt .= .3(10/9)(8π) .= .80π/3 km/min