An open box is to be constructed by cutting corners out of a 9in by 12in sheet of cardboard and folding up the sides. Find the dimensions which will maximize volume.

To determine the dimensions that will maximize the volume of the open box, we need to follow these steps:

1. Identify the variables: Let's assume the dimensions we need to determine are x and y. These will represent the length of the cut-out corners on the cardboard sheet.

2. Create an equation for the volume: The volume of the box can be determined by multiplying the dimensions together. Since we need to cut out squares from the corners, we need to subtract 2x (for both the length and width) from the original dimensions.

The volume equation will be: V = (9 - 2x)(12 - 2x)y

3. Simplify the equation: Expand the equation by multiplying the terms.

V = (108 - 42x + 4x^2)y

4. Maximize the volume: To find the maximum volume, we need to take the derivative of the equation with respect to either x or y, and set it equal to zero.

dV/dx = (-42 + 8x)y = 0

From this, we can determine that x = 5.25 inches.

5. Substitute the value of x to find y: Now that we have the value of x, substitute it back into the volume equation and solve for y.

V = (108 - 42(5.25) + 4(5.25)^2)y

Simplifying the equation, we get V = 157.5y - 11.38y^2

To maximize the volume, we can take the derivative of this equation with respect to y:

dV/dy = 157.5 - 22.76y = 0

From this, we can determine that y ≈ 6.92 inches.

6. Calculate the dimensions of the open box: Now that we have the values of x and y, we can subtract 2x from each side of the original dimensions to find the actual dimensions of the open box.

Length = 9 - 2x ≈ 9 - 2(5.25) ≈ 9 - 10.5 ≈ -1.5 inches (negative value doesn't make sense, so disregard it)

Width = 12 - 2x ≈ 12 - 2(5.25) ≈ 12 - 10.5 ≈ 1.5 inches (disregard the negative value as well)

Height = y ≈ 6.92 inches

Therefore, the dimensions that maximize the volume of the open box are approximately 1.5 inches (width), 6.92 inches (height), and disconsidering the negative value for the length.