December 2020
Problem of the Month

Wrapping Paper Problem
Submitted by Brijesh Dave



There is a rectangular sheet of wrapping paper whose dimensions are 32 centimetres by 20 centimetres.
Equal squares are cut out of each of the four corners of the paper.

Can you determine the maximum volume of a box which can be lined by suitably bending the wrapping paper to cover the base and sides of the box?

You must show your work to get credit.



Solution to the Problem:

The volume of the box is 1,152 cubic cm.

Let x = length of the side of one of the squares to be cut out.

Then the length and width of the base would be 32 - 2x and 20 - 2x.

The volume of the box would be:
V = (32 - 2x) (20 - 2x) x

Expanding, V = 4x3 - 104 x2 + 640x

At this point, you could graph the function to see where the greatest volume is.

Or you could use calculus to find the derivative and set it equal to zero.

dV/dx = 12x2 - 208x +640

Set equal to zero and divide by 4 to obtain: 3x2 -52x +160 = 0

Then factor: (3x - 40) (x - 4) = 0

So, x = 4 cm or x = 40/3 cm (not possible)

So V = 4 x 12 x 24 = 1,152 cubic cm.



Correctly solved by:

1. Brijesh Dave Mumbai City, Maharashtra, India
2. Colin (Yowie) Bowey Beechworth, Victoria, Australia
3. Veena Mg Bangalore, Karnataka, India
4. Ivy Joseph Pune, Maharashtra, India
5 Pearl Burruss Delta High School,
Delta, Colorado



Send any comments or questions to: David Pleacher