What is the volume in cubic centimetres?

Question

If 1300 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

What is the volume in cubic centimetres? 

Answer 1

Max Time Unlimited, Iterations Unlimited, Precision 0.000001, Use Automatic Scaling
Convergence 0.0001, Population Size 100, Random Seed 0, Derivatives Forward, Require Bounds
Max Subproblems Unlimited, Max Integer Sols Unlimited, Integer Tolerance 1%, Assume NonNegative


Cell Name Original Value Final Value
$F$3 L*L*H 1 4510.27



Cell Name Original Value Final Value Integer
$F$4 L 1 20.82
$F$5 H 1 10.41
The answer according to the optimizer tool is:

Base an square of side equal to 20.82 cm and the four sides four rectangles of

base 20.82 cm and height 10.41 cm with a total volume of 4510 cm³ Total area of used material 1300 cm²

Cell Name Cell Value Formula Status Slack
$F$6 L^2+4LH 1300 $F$6=1300 Binding 0

Answer 2

Again, it's useful to know that the box with the largest volume - to - surface area ratio is a cube. Your shape will be a cube (with a missing top).

 

Thus there will be five square faces with a total area of 1300 square centimeters. The area of each side will thus be:

 

1300/5 = 260

 

The length of each side will again be an irrational number (√260), because 260 is not a perfect square. This comes out to roughly:

 

16

 

The simplest way to do it from here would be to simply take the cube of your side length:

 

16³ = 4096

 

Your box's volume will be roughly 4096 cm³