Marquis de l'Hôpital Optimization Puzzle
Here's a classic geometry puzzle from the 1600s.
We have a beam, a hinged rod, a pulley, and a hanging weight.
How far down can the weight go?
The beam AB is 1 meter long and fixed horizontally.
At point A a rod AC is attached. The rod is 0.4 m long and can pivot freely.
At the end of the rod (point C) is a pulley.
A rope is tied at B, passes over the pulley at C, and then hangs straight down to a weight at D.
We also know:
- The rod AC has length 0.4 m
- The total rope length BC + CD is 1 m
As the rod swings, the pulley moves , and so does the weight.
What's the greatest possible depth that point D can reach below the beam?
This problem is attributed to the Marquis de l'Hôpital (1661–1704), author of one of the first calculus textbooks, Analyze des Infiniment Petits.
It is a beautiful example of how calculus can solve a very visual, physical problem.
Let's Set It Up
The rod can rotate, so we describe its position using an angle θ, measured from the horizontal beam.
- Point A is at (0, 0)
- Point B is at (1, 0)
- Point C is at (0.4 cos θ, 0.4 sin θ)
The rope has fixed total length:
BC + CD = 1
If we can find the length BC, then we can calculate CD.
Since the rope from C to D hangs vertically, the length CD tells us how far the weight hangs below point C.
So our plan is:
- Use geometry (the Distance Formula) to find BC in terms of θ
- Use the rope length to express CD in terms of θ
- Find the value of θ that makes CD as large as possible
In other words, we turn the geometry into a function, and then find its maximum.