What’s the greatest value of k for which you should use four posts rather than three?

Question

Quarantined in your apartment, you decide to entertain yourself by building a large pen for your pet hamster. To create the pen, you have several vertical posts, around which you will wrap a sheet of fabric. The sheet is 1 meter long — meaning the perimeter of your pen can be at most 1 meter — and weighs 1 kilogram, while each post weighs k kilograms.

Over the course of a typical day, your hamster gets bored and likes to change rooms in your apartment. That means you want your pen to be lightweight and easy to move between rooms. The total weight of the posts and the fabric you use should not exceed 1 kilogram.

For example, if k = 0.2, then you could make an equilateral triangle with a perimeter of 0.4 meters (since 0.4 meters of the sheet would weigh 0.4 kilograms), or you could make a square with perimeter of 0.2 meters. However, you couldn’t make a pentagon, since the weight of five posts would already hit the maximum and leave no room for the sheet.

You want to figure out the best shape in order to enclose the largest area possible. What’s the greatest value of k for which you should use four posts rather than three?

Answer 1

The pen will be either an equilateral triangle or a square.

In terms of the perimeter, P, the area of the triangle will be A = Sqrt(3) P²/ 36 ( from Heron formula)

In terms of the perimeter, P, the area of the square will be A = P² /16

The constraint for the triangle is 1 = 3 k + P and for the square 1 = 4 k + P

Thus the cross over value of k will satisfy

sqrt(3) (1 - 3k)² / 36 = (1 - 4k)² / 16

The solution to this is k = .08964

So for k < .08964 the square is better and for k > .08964 the triangle is better.

Answer 2

For the square, the weight of four posts will be 4k.

Thus, the perimeter will be (1 - 4k), so each side will be (1 - 4k)/4, and the area will be ((1 - 4k)/4)^2 =

(1/4-k)^2; k <= 1/4 (otherwise, a negative side length)

For the triangle, the weight of the three posts will be 3k, so the perimeter of the equilateral triangle is 1 - 3k, and the length of each side is (1-3k)/3. The area of an equilateral triangle is √3 s^2/4, so it will be

√3 ((1-3k)/3)^2/4 = √3(1/3-k)^2/4; k <= 1/3 (otherwise, a negative side length

Let's expand the two expressions.

1/16 - 1/2k + k^2

√3(1/36 - 1/6k + 1/4k^2)

Then, we need to find the k's such that 1/16 - 1/2k + k^2 >= √3(1/36 - 1/6k + 1/4k^2)

1/16 - √3/36 + (√3/6 - 1/2)/k + (1 - √3/4)k^2 >= 0

Note that 1/16 - √3/36 > 0 and 1 - √3/4 > 0, so we will either have 2 or 0 roots. However, if the value of k is greater than 1/4, this is not a valid solution.

Using the quadratic formula, k = 0.0896422471294551, 0.283073096072141

The difference will be greater than 0 for k < 0.0896422471294551 and k > 0.283073096072141, which is more than 1/4

You will have four posts rather than 3 if k < 0.0896422471294551