# Lighthouse problem: my answer does not match the key

Here is the problem:

A lighthouse is located in Lake Michigan, 300 feet from the nearest point on shore. The light rotates at a constant rate, making k complete revolutions per hour. At the moment that the beam hits a point on the shore 500 feet from the lighthouse, the point of light is traveling along the shoreline at a rate of 2,500 feet per minute. Find k.

          x
A___________B
|          /
|        /
300|      /500
|    /
|θ /
|/
L
(Lighthouse)


Here is my solution:

$$x = 500sin(\theta)$$ $$\frac{dx}{dt} = 500cos(\theta)\frac{d\theta}{dt}$$ When the light hits B, $\frac{dx}{dt} = 2500$ and $cos(\theta) = \frac35$.

Thus, $$2500 = 500\cdot\frac35\frac{d\theta}{dt}$$ $$\frac{d\theta}{dt} = \frac{25}3$$

But, the light makes k revolutions per hour, which we will convert to per minute. $$\frac{d\theta}{dt} = \frac{2\pi k}{60}$$

Now, we have $$k = \frac{250}{\pi}$$

But when I looked at the key, it was $\frac{90}\pi$. Where was I wrong?

• Hint: $x = 500 \sin(\theta)$ is correct at the given $\theta$, but $500$ is not a constant as $\theta$ varies, so you can not derive it against time as such. – dxiv Oct 3 '16 at 3:57

Your expression for $x$ is wrong for general $\theta$, since the 500 is not a constant. It should be $$x=300\tan\theta$$ Then $$\frac{dx}{dt}=300\sec^2\theta\cdot\frac{d\theta}{dt}=2500$$ Since $\cos\theta=\frac35$, $\sec^2\theta=\frac{25}9$. $$300\cdot\frac{25}9\cdot\frac{d\theta}{dt}=2500$$ $$\frac{d\theta}{dt}=3=\frac{2\pi k}{60}$$ $$k=\frac{90}{\pi}$$