Question

a box with a square base is taller than it is wide. in order to send the box through the US mail, the height of the box and the perimeter of the base cna sum to no more than 108 in. what is the maximum colume for such a box

Answer 1

The given box has the shape of a cuboid, since its height is greater than its width. Thus, the maximum volume for such box is 11200 $in^{3}$.

The volume of an object is a measure of it containing capacity. Since the given box has a taller height than its width, then it has the shape of a cuboid. The volume of a cuboid is given as:

volume = length x width x height

            = area x height

Given that the sum of the perimeter of its base and its height is not more than 108 inches, we can say; let the sides of the square base be represented by l and its height by h.

Then;

4l + h = 108

Therefore, maximum volume for the box can be attained when l = 20 inches and h = 28 inches.

So that;

4(20) + 28 = 80 + 28

                 = 108 inches

Thus;

maximum volume = area of the square base x height

                        = 400 x 28

maximum volume = 11200 $in^{3}$

The maximum volume for such a box would be 11200 $in^{3}$.

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