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This is more of math problem, but my doubt is about the measurement units of the final answer so I figured I'd post it here.

Problem:

A lighthouse is located on a small island 3 km away from a straight shoreline and its light makes 2 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 2 km away from the nearest point to the lighthouse.

Solution:

Let $x$ be distance between the beam and the nearest point and $\theta$ the angle between the line that goes from the lighthouse to nearest point and the one that goes to where the beam is on the shoreline. We know that $\frac{d\theta}{dt} = \frac{\pi}{30}$ radians per second and that $\tan(\theta)=\frac{x}{3}$. We need to find $\frac{dx}{dt}$. Thus

\begin{align*} x &= 3\tan(\theta)\\ \frac{dx}{dt} &= 3\frac{d}{dt}\tan(\theta)\\ &= 3\sec^2(\theta)\frac{d\theta}{dt}\\ &= 3(\tan^2(\theta)+1)\frac{\pi}{30}\\ &= (\frac{x^2}{9}+1)\frac{\pi}{10}\\ \end{align*}

And if we plug for $x=2$ we get $\frac{13\pi}{90}$.

Question:

What is the measurement units of the answer? I thought it should be kilometers per second, because it is only type of answer that makes sense, but how does it follow from the steps in the solution?

WeakestTopology
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2 Answers2

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HINT: this is why it's always good to carry the units along throughout your derivation. Try starting the derivation by writing $$ x = (3 \text{ km}) \tan(\theta) $$ and note that the units of $d \theta/dt$ when you plug it in are radians per second, which you can write as $\text{s}^{-1}$. (If you're unclear on why this is, see this question.) Once you've done this, the way the units cancel & combine should become clearer.

Michael Seifert
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Your question could be asked like this. I have position as a function of time, $y = x(t)$, where x is in meters, and t in seconds. I can take the derivative to get velocity, $v = dy/dx$. How do I get the right units? If $y = 3t$, I get $v = 3$, not $v = 3 m/s$

The answer is in the definition of the derivative. It is the limit of $\Delta x/\Delta t$ as $\Delta t \to 0$. So the units come out the same as if you had divided.

mmesser314
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