Question

A piece of cardboard measuring 8 inches by 11 inches is formed into an open-top box by cutting squares with side length x from each corner and folding up the sides. Find a formula for the volume of the box in terms of x.Find the value for x that will maximize the volume of the box.

Answer 1

The formula for the volume of the box in terms of x is (8 - 2x)(11 - 2x)(x). The value for x that will maximize the volume can be found by taking the derivative, setting it to zero, and using the second derivative test.

Step-by-step explanation:

To find the formula for the volume of the box in terms of x, we need to consider the dimensions of the box. After cutting squares with side length x from each corner, the length of the box becomes 8 - 2x and the width becomes 11 - 2x. The height of the box will be x. So, the formula for the volume will be V = (8 - 2x)(11 - 2x)(x).

To find the value for x that will maximize the volume of the box, we can take the derivative of the volume formula with respect to x, set it to zero, and solve for x. After finding the critical points, we can use the second derivative test to determine which point gives a maximum volume.

Answer 2

The formula for the volume of the box and the value of x that will maximize the volume of the box are;

The volume of the box, is; V = 4·x³ - 38·x² + 88·x

The value of x that will maximize the volume of the box is; 3 1/6 inches

The steps used to analyze the volume of the box are as follows;

Let x represent the side length of the square;

The height of the open box = x

The length of the open box = 11 - 2·x

The width of the open box = 8 - 2·x

Volume of the box, V, is; x × (8 - 2·x) × (11 - 2·x) = 4·x³ - 38·x² + 88·x

The maximum volume can be found from the rate of change of the function for the volume

The maximum volume is the point where the rate of change of the volume function is zero, dV/dx = 0, therefore;

V = 4·x³ - 38·x² + 88·x

dV/dx = 12·x² - 76·x + 88

Therefore, at the x-value for the maximum volume, we get;

dV/dx = 0

12·x² - 76·x + 88 = 0

x = -(-76)/(2 × 12)

x = 19/6

19/6 = 3 1/6

The value of x that will maximize the volume of the box is x = 3 1/6 inches