Optimum volume and surface area

Lina is thinking about how to make cans and at the same time use as little metal as possible. That would be optimal. The can she holds in her hand holds 330 milliliters. In order for it to be easy to hold, the can has the shape of a cylinder. Cylinder volume is calculated by multiplying the base area, which is a circle, times the height of the cylinder.

So, pi times radius squared which is the circle, times the height. The can can hold 330 milliliters, or 330 cubic centimeters. That is the constraint on the volume. Therefore the constraint equation is, pi times radius squared times the height, which equals 330. We calculate the amount of material required by adding up the three areas of the can, the lid, the bottom and the curved surface around the can.

This is the total surface area for a cylinder. The lid and bottom are both circles in shape. The area of the lid is pi times radius squared. But we also have a bottom, so we have to multiply the area by two. We make a cut along the height of the curved surface around the can and flatten it.

That surface has the shape of a rectangle. The width of the rectangle is the perimeter of the can. The height of the rectangle is the height of the can. The area for this surface is the width times the height or pi times two times the radius, times the height. Add the areas for the lid, the bottom and the rectangle, and we get the total surface area.

Lina wants to use as little material as possible. She wants to optimize the area. Therefore, she must find out which values of the radius, r, and the height, h which give the smallest area. This means that the formula for calculating the total surface area is an optimization equation. It can be simplified so that it contains only one unknown value.

Let’s test replacing h in the optimization equation by using the constraint equation. Divide by 'pi and radius squared' on both sides of the equals sign. Then h, height, equals: 330 divided by 'pi times the radius squared'. Now we replace h in the optimization equation with '330 divided by pi times the radius squared' The new total surface area formula, becomes 'pi times radius squared, times two, plus 660 divided by the radius'. Pause the movie and make sure it is correct.

How do we now get the smallest, the optimal area? Let’s test different values of r with a table. Enter some values for the radius, r, in column A. And in column B, write the different values you get. As the radius increases, the total surface area decreases, until the radius has increased to 5, then the area increases again.

So, where the radius is about 4 centimeters, that is the smallest total surface area. But it is not certain that it is exactly 4 centimeters that gives the smallest total surface area. With the radius 3,9 [3.9] we get a smaller total surface area. We test with 3,8...3,7...3,6...! [We test with 3.8...3.7...3.6...!] Aha! The area becomes smaller until the radius is reduced to 3,7.[3.7] Let’s try with two decimals.

Then the smallest area is when the radius is 3,74 [3.74]. You can continue with three and four and five decimals, but we will stop here. If the radius is 3,74 [3.74] centimeters, the height is about 7,5 [7.5] centimeters. A radius of 3,74 [3.74] centimeters means that the diameter is 7,48 [7.48] centimeters. We round this of to 7,5. [7.5] The diameter is thus about the same length as the height.

The minimum total surface area of a cylinder appears to be when the height is equal to the diameter. So that is the way a can shaped like a cylinder should be made, to consume minimal material. This is the optimal result for Lina. But the height of the can Lina has is greater than the diameter. This is probably because it is easier to carry and drink from such a can.

Oh Lina ... please / seriously.