Question

How would you change the dimensions of a rectangular prism box so that it holds more and uses less cardboard to make?

Answer 1

You make it a cube.

Explanation:

It has been asked that "how would you change the dimensions of a rectangular prism box so that it holds more and uses less cardboard to make? This means we increase the volume and reduce surface area.

We do not normally do this way. What we do is to find that given a surface area of a rectangular prism, how can we maximize its volume by changing dimensions. If we fix te surface as `A`, then `A=lw+wh+lh`, where `l` is length of prism, `w` is its width and `h` is its height.

As number of variables are involved, it is a bit complicated and we use partial differential calculus due to involvement of multiple variables. A detailed proof is available here, whose net result is that volume is maximized when all dimensions of rectangular prism are same i.e.

you make it a cube and given `A`, its dimensions are `sqrt(A/6)` and volume is `(sqrt(A/6))^3`