# Light in a Lighthouse

In summary, the problem involves finding the speed of a spot of light along the shore when it is 2 miles away from the point of the shore nearest the light. Using the equation \frac{dx}{dt} = \frac{dx}{d\theta} \frac{d\theta}{dt}, we can solve for the speed by constructing a circle with radius 2 around the lighthouse and finding the midpoint of a chord that is 1 mile away from the lighthouse. The final answer is \boxed{10\pi}.

## Homework Statement

A light in a revolving lighthouse located 1 mile away from a straight shoreline turns at 1 revolution per minute. Find the speed of the spot of light along the shore when it is 2 miles away from the point of shore nearest the light.

## Homework Equations

$\frac{dx}{dt} = \frac{dx}{d\theta} \frac{d\theta}{dt}$

## The Attempt at a Solution

Here is my solution, although I don't know if it (and the answer it produces) is correct or not

Let the lighthouse be point O. Construct a circle with radius 2 around O, and a chord such that the distance from O to the midpoint of the chord is 1. Let the midpoint of the chord be point M, and one of the intersections of the chord with the circle be point N. Now let half the length of the chord be x.

We have $\tan{\theta} = x$, so $\frac{dx}{d\theta} = \sec^2{\theta}$. This also equals $1 + \tan^2{\theta}$, or $1+x^2$. Since x=2, it's basicaaly 5.

Now $\frac{dx}{dt} = \frac{dx}{d\theta} \frac{d\theta}{dt}$. $\frac{dx}{d\theta}$ is $\sec^2{\theta}$, and $\frac{\text{d} \theta}{\text{dt}}$ is $2\pi$, so the answer is $\boxed{10\pi}$.

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Mod note: I removed all of the \text stuff in your LaTeX. Now it renders correctly.