SOLUTION: A company needs to create an open topped box to shipts its product. The box is going to be made by cutting congruent squares from each corner of a piece of cardboard, and then fol

Question 112912: A company needs to create an open topped box to shipts its product. The box is going to be made by cutting congruent squares from each corner of a piece of cardboard, and then folding the carboard to create a container of maximum volume. What is the size of each square that is removed from the corners, what are the dimensions of the box, and what is the volume of the box with the largest area? The piece of cardboard has a length of 65 inches and a width of 56 inches.
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A company needs to create an open topped box to ships it's product. The box is going to be made by cutting congruent squares from each corner of a piece of cardboard, and then folding the cardboard to create a container of maximum volume. What is the size of each square that is removed from the corners, what are the dimensions of the box, and what is the volume of the box with the largest area? The piece of cardboard has a length of 65 inches and a width of 56 inches.
:
Draw a rough diagram of the rectangular piece of cardboard, label it 65 by 56
Let the side of the small squares = x
It will be apparent that the base dimensions of the box will = (65-2x by (56-2x)
The height will = x
:
let y = volume of the box:
y = x * (65-2x) * (56 - 2x)
y = x(4x^2 - 242x + 3640)
y = 4x^3 - 242x^2 + 3640x
:
The easiest way to determine max volume is graph this equation

:
Note that max volume seems to occur when x = 10 inches
:
To answer the questions then:
The size of the squares are 10" by 10"
The volume:
(65-20)*(56-20) * 10
45 * 36 * 10 = 16200 cu inches
:
You can confirm this by finding the volume using x = 11 and x = 9
The volume will be less.
:
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