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A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides. Express the volume V of the box as a function of x.
Solution:
To visualize the problem, let's draw the figure first as follows:
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| Photo by Math Principles in Everyday Life |
If you cut and remove the four squares of side x and then fold up the four sides, the figure is a rectangular parallelepiped.
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| Photo by Math Principles in Everyday Life |
The volume of a rectangular parallelepipied is given by the formula
V = L W H
Substitute the given dimensions, we have
V = (20 - 2x)(12 - 2x)(x)
V = [2(10 - x)][2(6 - x)](x)
V = 4x (10 - x)(6 - x)
Therefore,
V(x) = 4x (10 - x)(6 - x)
where 0 < x < 6