# LessonRelated Rates: Lighthouse Beam

#### MarkFL

##### La Villa Strangiato
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Math Helper
A lighthouse is fixed $$y$$ units from a straight shoreline. A spotlight in the lighthouse revolves at a rate of $$R$$ revolutions per minute, shining a spot along the shoreline as it spins. At what rate is the spot moving when it is along the shoreline $$x$$ units from the shoreline point closest to the lighthouse?

Consider the following diagram:

As we can see, we may state:

$$\displaystyle \tan(\theta)=\frac{x}{y}$$

Now, let's differentiate with respect to time $$t[$$, bearing in mind that while $$x$$ is a function of $$t$$, $$y$$ is a constant.

$$\displaystyle \sec^2(\theta)\cdot\d{\theta}{t}=\frac{1}{y} \cdot\d{x}{t}$$

Since we are being asked to find the speed of the spot, whose position is $$x$$, we want to solve for $$\displaystyle \d{x}{t}$$:

$$\displaystyle \d{x}{t}=y\sec^2(\theta) \cdot\d{\theta}{t}$$

Let's let the angular velocity be given by:

$$\displaystyle \omega=\d{\theta}{t}$$

and from the diagram and the Pythagorean theorem, we find:

$$\displaystyle \sec^2(\theta)=\frac{x^2+y^2}{y^2}$$

Hence, we have:

$$\displaystyle \d{x}{t}=\frac{\omega}{y}(y^2+x^2)$$

Now, the angular velocity is in radians per unit of time, and since there are $$2\pi$$ radians per revolution, we may write:

$$\displaystyle \omega=2\pi R$$

Hence:

$$\displaystyle \d{x}{t}=\frac{2\pi R}{y}(y^2+x^2)$$

anemone and harpazo

Amazing!