Finding the largest possible area

Question

Suppose you want to build a box,and you have 100 square feet of cardboard material to use.You want the box to have as much volume as possible.Find the whole number dimensions that will produce the greatest volume.What is this volume?What is this volume? Show all calculations.

Answer 1

Assuming your box is a perfect cube

You can divide the surface area by 6 for the 6 equal square faces of your box

100/6 = 16.67 ft² per side of the box

√16.67= 4.08 ft maximum per side (but this is not a whole number)

4 x 4 x 4 cube works with each face being 4 x 4 squares

6(4 x 4) = 6(16) = 96 ft²

It could hold a volume of

4³ = 64 ft³

If your box is rectangular 7 x 4 x 2 works for a surface area of 100 ft² with a volume of 56 ft³

(7 x 4)2 = 56 ft²

(7 x 2)2 = 28 ft²

(4 x 2)2 = 16 ft²

56+ 16 + 28 = 100 ft²

You want to explore volume to surface area ratios approaching 1.

A closed surface area has to spread around and over a volume to contain it so surface area is usually larger.

I hope you find this information useful