Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 13 in. by 8 in.

1) To make a rectangular box you need to cut squares from the four corners of the rectangular sheet.

2) Call x the length of the sides of the squares cut off.

3) The base of the box will have dimensions: (13 - 2x) and (8 - 2x)

4) The height of the box will be x

5) The volume of the box will be the area of the base times the height:

Volume = (13 - 2x)(8 -2x)x = (4x^2 - 42x + 104)x = 4x^3 - 42x^2 + 104x

6) The maximum volume is calculated by finding the point where the derivative of the volume is zero =>

d (volume) / dx = 12x^2 - 84x + 104 = 0

7) Solve the quadratic equation 12x^2 - 84x + 104 = 0

=> 4(3x^2 - 21x + 26) = 0

=> 3x^2 - 21x + 26 = 0

=> 3 (x^2 - 7x) + 26 = 0

=> 3 [(x - 7/2)^2 - (7/2)^2] + 26 = 0

=> 3 (x - 7/2)^2 - 3* 49/4 + 26 = 0

=> 3 (x - 7/2)^2 = 3*49/4 - 26

=> (x -7/2)^2 = (49/4 - 26/3)

=> x = 7/2 +/- √(49/4 - 26/3)

x = 7/2 + √3.583 and x = 7/2 - √3.583

x = 5.393 and  x = 1.607

=> Volume =

1) 4(5.393)^3 - 42(5.393)^2 + 104(5.393) = -33.26 ---> it does not have physical meaning

2) 4(1.607)^3 - 42(1.607)^2 + 104(1.607) = 75.27 ---> this is the answer

75.27 in^3