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A box with a square base and an open top is being constructed out of a cm2 of material. If the volume of the box is to be maximized, what should the side length of the base be? what should the height of the box be? what is the maximal volume of the box? your answers should be in terms of a.

Answer :

The maximum volume of the box in terms of A will be √3A/4 -3√3/4 cu cm.

It is given that the box has an open top and a square base.

This implies length = breadth.

Let them be x cm

Now, the area of the box is A sq cm

This means

lb + 2bh + 2lh = A

or, x² + 2xh + 2xh = A

or, 4xh + x² = A

or, h = A/4x - x/4

Now volume (V) = lbh

= x²(A/4x - x/4)

= Ax/4 - x³/4

Now to maximize volume we will first differentiate with respect to x and equate it to zero. Hence we get

V' =0

or, A/4 - 3x²/4 = 0

or 3x²/4 = A/4

or, x² = 3

or, x = √3

Now t check whether this gives us the maximum volume we will calculate V"(√3)

V" = -3x/2

V"(√3) = -3√3/2

Since V"(√3) is negative we will get maximum value for x = √3

Now the volume is

A√3/4 - (√3)³/4

= √3A/4 -3√3/4 cu cm

To know more about maximization visit

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