Determine the dimensions of the cut-out that will produce the box of maximum volume. What are the dimensions of the box with maximum volume?

Question

A square of side x inches is cut out of each corner of a 10in x 18in piece of cardboard and the sides are folded up to form an open-top box.

Answer 1

Cutting out a square corner with sides of x would give the box sides of 18 – 2x and 10 – 2x and a height of x. We can make an equation to find the volume of the box that looks like:

V = (18 – 2x)(10 – 2x)(x) = 4x³ – 72x² + 180x

By setting the equation equal to 0, we can find the roots of the equation:

4x³ – 72x² + 180x = 0

After solving, we find that we have 3 roots at:

x = 0

x = 3

x = 15

Because of the shape of the polynomial function, we know that we have a local maximum between x = 0 and x = 3. We can find this maximum by taking the derivative of the equation:

d/dx (4x³ – 72x² + 180x) = 12x² – 144x + 180

Solving the quadratic equation to find the roots of the equation gets us:

x = 6 ± sqrt(21)

Since x = 6 – sqrt(21) ≅ 1.42 is the one of these values in the range x = 0 and x = 3, this is our value for x that maximizes the volume of the box.

Therefore the cutout dimensions that maximize the volume of the box is ~1.42in