Maximum Volume

Question

What is the maximum volume in cubic inches of an open box to be made from a 10-inch by 20-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? Your work must include a statement of the function and its derivative. Give one decimal place in your final answer. (10 points)

Answer 1

Hi Gail, the volume of a box is: V = L * W * H. We are starting with a piece of cardboard 10 inches by 20 inches. If we let x represent the height of the box, x will also be the side of the square taken out of each corner.

 

The length of the box becomes 20 - 2x. The width of the box becomes 10 - 2x. The height of the box is x.

 

Substituting in the equation for the volume gives:

 

V = x * (20 - 2x) * (10 - 2x)

 

Multiply the right side to get:

 

V(x) = 4x³ - 60x² + 200x

 

Take the first derivative of V:

 

V'(x) = 12x² - 120x + 200

 

To find x, set the first derivative equal to 0:

 

12x² - 120x + 200 = 0

 

Use the quadratic formula to solve for x:

 

a = 12

 

b = -120

 

c = 200

 

x = (120 ± √((-120)² - 4 * 12 * 200) / (2 * 12)

 

Use the positive value first:

 

x = (120 + √(14400 - 9600) / 24

 

x = (120 + 69.3) / 24

 

x = 189.3 / 24

 

x = 7.9 inches

 

This solution can be discarded because it would make the width a negative number.

 

Use the negative value next:

x = (120 - √(14400 - 9600) / 24

x = (120 - 69.3) / 24

x = 50.7 / 24

x = 2.1 inches

 

Take the second derivative of V to check for a local maximum.

 

V''(x) = 24x - 120

 

Substitute for x:

 

V''(2.1) = 24 * 2.1 - 120

 

The second derivative is less than zero, x is a maximum.

 

Sides of the box:

 

Length = 20 - 2 * 2.1 = 17.8 inches

 

Width = 10 - 2 * 2.1 = 7.8 inches

 

Height = 2.1 inches

 

Questions?