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An open-topped box is to be made from a rectangular piece of material, 8 by 10 inches in size, by cutting equal squares from each corner and turning up the sides. Find the maximum volume enclosed by a box made in this way.
We need to express the volume of the box in terms of X, the unknown size of the squares removed. After the sides are folded up, the height of the box will be X. Its reduced length is 10 – 2X and its reduced width is 8 – 2X. Multiplying these together will give us the volume in terms of X:
We can find the value of X which maximizes this by differentiating V and setting the derivative to 0:
We can use the quadratic formula to solve this:
inches
(Here we have rejected the value 4.53 inches as being meaningless compared to the dimensions of the original sheet.)
We can check this result by calculating the volume with this value and with nearby values, substituted in the equation for V above:
|
X |
1.4 |
1.47 |
1.6 |
|
Volume |
52.4 |
52.5 |
52.2 |