Determine the maximum volume that can be created by cutting the 17.5 cm by 18.75 cm cardboard into an open box.

Question

What would be the maximum volume?

Answer 1

Assumption

We cut out a shape that is a cross. The center is a rectangle with dimensions equal to the length and width (L and H) of the box. (This will be the bottom of the box)

On each side of the center rectangle is another rectangle which is either L or H times the height of the box (H)

(These will be the 4 sides of the box)

So the dimensions of the cross are L + 2H, and W + 2H

Using as much of the cardboard shape as possible, we have:

L + 2H = 17.5

W + 2H = 18.75

Write both in terms of H:

L = 17.5 - 2H

W= 18.75 - 2H

Now we can write an equation for the volume:

H*L*W = H * (17.5 - 2H) * (18.75 - 2H)

Expand this and simplify, and you will have a 3rd-order polynomial

To find the maximum, take the 1st derivative and set it equal to zero. This will give you two values for H. Sketch the graph of the function to determine which is the maximum and minimum volume.

Answer 2

The area of the cardboard = 17.5 x 18.75 = 328.125 cm² ( I am going to assume that we can manipulate 328.125 cm² in any box dimension that we want but be sure to keep 328.125 cm² of material only ?!)

We know from previous max-min problems, that the maximum area is achieved when length = width

We also know that the maximum volume is attained when the length = width = height

Lets look at the surface area for the open top box is the dimensions are denoted by

x = width

y = length

z = height So, the area of the open top box is 2xz + 2yz + xy = 328.125

But all dimensions are equal so let x be the lone variable so that Area = 2x² + 2x² + x² = 328.125

Collecting terms, we have Area = 5x² = 328.125 or x = 8.1 cm

So, the maximum volume is V = xyz = x³ = 531.44 cm³

Answer 3

Assuming that you are cutting a square of length x from each corner of the piece of cardboard, and then folding up the flaps to form an open box, the dimensions of the box are 17.5 - 2x by 18.75 - 2x by x.

V = volume = x(17.5 - 2x)(18.75 - 2x) = x(4x² -72.5x + 328.125)

V = 4x³ - 72.5x² + 328.125x, where 0 < x < 8.75 (Note: 8.75 s half of 17.5)

dV/dx = 12x² - 145x + 328.125

dV/dx = 0 when x = [145 ± √5275] / 24 = 9,07 or 3.02

Since x can't be larger than 8.75, we can eliminate 9.07.

When 0 < x < 3.02, dV/dx > 0, so V is increasing

When 3.02 < x < 8.75, dV/dx < 0, so V is decreasing

There is a relative (and absolute) max when x = 3.02.

Max volume = 4(3.02)³ - 72.5(3.02)² + 328.125(3.02) = 439.88 cm³