If an open box has a square base and a volume of 500 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction.

Question

If an open box has a square base and a volume of 500  and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction.

Quizplus · Accepted Answer

Let the length of the side of the square base be x and the height of the box be h. Then, from the given information, we have:

x^2h = 500

To minimize the amount of material used, we want to minimize the surface area of the box. The surface area consists of the area of the base (x^2) and the area of the four sides (4xh). So, we want to minimize:

A = x^2 + 4xh

Using the equation from above, we can solve for h in terms of x:

h = 500/x^2

Substituting this into the expression for A, we get:

A = x^2 + 4x(500/x^2)
A = x^2 + 2000/x

To minimize A, we can take the derivative and set it equal to zero:

dA/dx = 2x - 2000/x^2 = 0
2x = 2000/x^2
x^3 = 1000
x = 10∛10

Substituting this back into the equation for h, we get:

h = 500/(10∛10)^2
h = 50/(∛10)^2

Therefore, the dimensions of the box are approximately x = 10∛10 and h = 50/(∛10)^2. Rounded to three decimal places, this is x ≈ 11.829 and h ≈ 19.245. Therefore, choice B is the correct answer.

If an Open Box Has a Square Base and a Volume