Math quick question

You have 500 cm² of material available to cover different shapes. what is the maximum volume of each object that you could make with this amount of material?
A.) a cube
B.) a sphere

I've been stuck on this for so long, please show me how to do this step by step, thanks :)

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1. So you're given the limitation of 500 cm^2. This is essentially the surface area of the different solids if you choose to form them into that shape.

The best way to calculate the maximum volume with a fixed surface area is to find the ratio of volume to surface area. The solid with a larger ratio is is able to hold more volume per surface area, which naturally means it can hold more volume with a fixed surface rea.

First the cube:

Surface area of the cube, you need to add the area of all 6 sides. For simplicity sakes, we assume that the length of the cube is variable x. So the surface area is equal to 6*(x^2). The volume is much easier calculate, x^3.

The ratio between volume to surface area of a cube is x^3/(6*x^2). Cancel out a x^2 from the top and bottom and you get x/6.

Now the sphere:

Once again, we assume for simplicity sakes that the length of the radius is x. If you still don't understand this, the reason we do this is because if we choose to set a ratio comparing the ratios of sphere and cubes, the x variable cancels.

The surface area for a sphere is 4πx^2. The volume is 4/3πx^3. Set a ratio of volume to surface area. 4/3πx^3/4πx^2 which simplifies to x/3.

Now compare the different ratios.
x/6 and x/3. As you can see, the ratio of volume to surface area is much bigger for the sphere than the cube. Therefore forming a sphere is the best way to maximize volume. It's the same logic with 2D shapes. A circle can hold more area with a set perimeter than a square.

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