Lighthouse Related Rates | Dr. Loveless Curiosity Lab

Student Video Explanations

Start with the 2D geometry picture, then see how the same idea can be viewed in 3D.

2D YouTube Tutorial

3D YouTube Tutorial

Lighthouse Showcase Visual

A nighttime lighthouse scene with rotating beam, shoreline, hillside, and moving water.

This 3D scene is here mostly for fun. It will not help you solve the related-rates problem, but it captures the setting: the lighthouse, the sweeping beam, the shoreline, and the ocean at night.

Given

Fixed Distance The lighthouse is a distance \(a\) from a straight shoreline.

Angle Let \(\theta\) be the angle the beam makes with the line perpendicular to the shore.

Shore Position Let \(x\) be the distance of the light spot along the shore from the closest point to the lighthouse.

Want

Find the speed of the light spot along the shoreline.

\(\displaystyle \tan(\theta)=\frac{x}{a}\)

so the geometry can be written as

\(\displaystyle x=a\tan(\theta)\)

Results

Differentiate with respect to time:

\(\displaystyle \frac{dx}{dt}=a\sec^2(\theta)\frac{d\theta}{dt}\)

A constant angular speed can create a very non-constant shoreline speed.

Where It Started

This is one of the classic related-rates problems from Math 124. Students often understand the picture of the lighthouse, but still find it difficult to turn that picture into an equation before differentiating.

That makes it a great showcase problem: simple to describe, highly visual, and rich enough to revisit with better explanations and better pictures.

What to Notice

The lighthouse rotates smoothly, but the bright spot on the shore does not move smoothly at all. It may move slowly at first and then race down the shoreline.

That is the heart of the problem: constant rotational motion can create dramatically changing linear motion somewhere else.

Student Video Explanations

Where It Started

What to Notice

Further Reading