An open box is to be made from a square piece of cardboard having dimensions 20 inches by 20 inches by cutting out squares of area from each corner as shown in the figure below. Express the volume V of the box as a function of x .

Question

An open box is to be made from a square piece of cardboard having dimensions 20 inches by 20 inches by cutting out squares of area  from each corner as shown in the figure below. Express the volume V of the box as a function of x .

Quizplus · Accepted Answer

Let x be the length of the square that will be cut out from each corner. 
The height of the box will be x inches. 
The length of the sides of the base (after the corners are cut out) will be (20 - 2x) inches.

Therefore, the volume of the box is:

V = height x length x width

V = x(20 - 2x)(20 - 2x)

V = 4x(10 - x)(10 - x)

Expanding:

V = 4x(100 - 20x - 20x + x^2)

V = 4x(x^2 - 40x + 100)

V = 4x^3 - 160x^2 + 400x

Taking the derivative to find critical points:

V' = 12x^2 - 320x + 400

Setting V' = 0:

12x^2 - 320x + 400 = 0

3x^2 - 80x + 100 = 0

Using the quadratic formula:

x = [-(-80) ± sqrt((-80)^2 - 4(3)(100))]/(2(3))

x = [80 ± sqrt(6400 - 1200)]/6

x = [80 ± sqrt(5200)]/6

x ≈ 3.69 or x ≈ 22.31

Since the length of the square that can be cut out at most is 10 to evenly cut the corner of the box, we can discard x ≈ 22.31.

To determine whether x ≈ 3.69 is a maximum or minimum volume, we can test a value less than and greater than this value.

If x = 3:

V = 4(3)^3 - 160(3)^2 + 400(3) ≈ 372

If x = 4:

V = 4(4)^3 - 160(4)^2 + 400(4) = 448

Therefore, the volume is maximized when x ≈ 4.

Substituting x = 4 in the expression for V:

V = 4(4)^3 - 160(4)^2 + 400(4) = 448

Therefore, the volume of the box is a maximum when x = 4 inches.

An Open Box Is to Be Made from a Square