The Mirror Trick

A trough with trapezoidal cross section is formed by turning up the edges of a 30 inch wide sheet of aluminum. So the bottom base I have labeled as 30-2x where each x is the length of the turned up pieces. What I need to do is find the cross section with the maximum amount of area.

(Remark. This problem can be solved by calculus, although it is somewhat lengthy because you have to maximize with respect to two variables simultaneously (namely the angle to bend and the length of the bent pieces). In this article we will investigate an elementary geometric technique that can be used to solve this and many other problems. If you can't think of a geometrical solution, work it out using calculus, then look at your final answer and think whether there might be a geometrical method to get the same answer.)

Need a hint? Click here.

Click here for the complete solution.

Hint 1

Draw a picture of the trough, then draw a mirror image reflected across the open top of the trough.

This gives you two pieces of aluminum forming a closed pipe. The perimeter of the pipe's cross-section is twice the width of the aluminum sheet, namely 60 inches, and the area is twice the area of the trough's cross-section area. Therefore whatever shape gives the maximum trough size also gives the maximum pipe size, and vice versa. Furthermore the pipe cross-section is a six-sided figure. Now it`s easy to finish: Of all six-sided figures with a given perimeter, which shape has the largest area? Therefore what?

The Rest of the Solution

It's a fact of geometry that, of all n-gons with a given perimeter, the regular n-gon has the largest area. A familiar example is the 4-gon or quadrilateral; of all quadrilaterals with a given perimeter, the square has the largest area.

Therefore our pipe has the largest cross-section when it is a regular hexagon, and the trough has the largest cross-section when it is half of a regular hexagon. A regular hexagon has six equal sides, and six equal interior angles of 120 degrees each, so the answer for our trough is that x = 10 and the interior angle A = 120 degrees.

The Mirror Trick and Optimizing Without Calculus

This idea is called the Mirror Trick. This is probably not a standard term; I learned the trick and the term from Richard J. Stanley of the University of California at Berkeley.

In many cases an apparently complicated optimization can be reduced to a simple problem by forming a mirror image. Below are some more examples for you to work on.

Ivan Niven wrote a book, Maxima and Minima Without Calculus, showing how to solve many optimization problems without calculus. He calls the Mirror Trick the "reflection principle" and gives some more examples there (section 3.6, "The Reflection Principle", pp. 63-67).

Largest Rectangular Pen

This is a familiar calculus exercise that can be done with the Mirror Trick. Suppose we already have a fence, and we want to build a rectangular pen along side it, of maximum area, using one side of the fence as a side of the pen.

We have 100 feet of fencing to make the pen. What size should we make it?

Largest Triangular Pen

This is a similar problem that we received at MathNerds. Suppose we want instead to build a right triangular pen of maximum area against an existing fence.

We have 300 meters of fencing to make the pen. What size should we make it?

Shortest Path To The Fire

This one is in many calculus books. Suppose we need to run to a fire with a bucket of water to put it out, but first we have to run to the river with the bucket to fill it. What's the shortest path? (Hint: reflect the fire across the river.)

There's a physics problem about real mirrors that uses the same picture and solution: If you want to shine a beam of light at a mirror to reflect off the mirror and hit some specified point, where would you aim it? (Physics says that light follows the shortest path.)

Billiard Ball Problem

(George Pólya) The mirror trick can be used for other types of problems too. Suppose we place a ball on a billiard table at some arbitrary position. We wish to drive the ball in such a way that it will bounce off the four sides of the table and the pass through its original position. What direction should we drive the ball?

References

• You can view Dan's original question here.
• We received the triangular pen as a question at MathNerds; you can read it here.
• The billiard ball problem comes from: George Pólya, Induction and Analogy in Mathematics (volume 1 of Mathematics and Plausible Reasoning ), Princeton University Press, 1954, problem 9.13.
• An online proof of the fact that the regular n-gon has the largest area is at the Math Forum. A printed proof is in: Richard Courant, Herbert Robbins, and Ian Stewart, What is Mathematics? 2nd edition, Oxford University Press, 1996, pp. 373-376.
• Ivan Niven, Maxima and Minima Without Calculus, Mathematical Association of America, 1981.