If an open box has a square base and a volume of 91 in.3

Question

If an open box has a square base and a volume of 91 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (Round your answers to two decimal places.)

Answer 1

Nathalie M.'s answer is correct, and here is the explanation:

Let the box have a square base, a in on each side. Let the height of the box be b. Then:

Area = A = a^2 + 4*a*b

Volume = V = a^2 * b

Substituting we can rewrite the Area:

A = x^2 + 4*V/a

Then to find the minimum we take the derivative:

dA/da = 2a - 4V/a^2

Setting the derivative equal to 0 to find the minimum gives:

4V = 2a^3 or a = 5.67 in. The corresponding height is 2.83 in.

Answer 2

The dimensions of the box are as follows:

the base dimensions are x=5.67 & y=5.67 & the height of the box z= 2.83 inches.

The box surface area is xy + 2xz + 2yz = 91 in³. The volume of the box is xyz (with open top).

The answer is obtained by using Lagrange Multipliers and setting x=y.

If you need more details, I will gladly try to forward details. Since volume is base area x height & we know that the minimum surface area with area xy is found when x=y. Then x was equated to y in the Lagrange Multiplier equations in order to make the math manageable.