Question

A box is to be made out of a piece of cardboard 38 cm x 38 cm, by cutting equal squares out of the

corners and turning up the sides. Find the volume of the largest box that can be made.

Answer 1

I assume that this box does not need a lid.

Suppose we cut a square of side x out each corner of the piece of cardboard, and then turning up the sides. The dimensions of the resulting box will be:

height = x

width = 38–2x

length = 38–2x

So volume  = V = x (38–2x) (38–2x)

Multiplying out the brackets gives V = -4x3 - 1444x

We need to find the value of x that maximises V. We can do this by differentiation:

dV/dx = -12x2 + 1444

lat the maximum, dV/dx = 0, -12x2 + 1444 = 0

Dividing both sides of this equation by 4 gives -3x2 + 361= 0

Factorising gives 

$x=+-\sqrt\frac{361}{3}$

so, $x_1 \approx 10,97$ or $x_2 \approx -10,97$

x=10,97 gives $V=10,97(38-10,97)^2 \approx8014,91$ =6859

x=-10,97 gives $V=-10,97(38+10,97)^2 \approx -26306,72$ which is a silly answer.

So the answer is that the maximum volume is 8014 cubic inches, and this is achieved by cutting a square of side 3 inches from each corner.