Question

You are tasked with constructing a rectangular box with a square base, an open top, and a volume of 184 in3. Determine what the dimensions of the box should be to minimize the surface area of the box. What is the minimum surface area? Keep your answer in radical form and omit units.

Answer 1

To minimize the surface area of the rectangular box with a square base and open top, we can solve equations for volume and surface area.

Step-by-step explanation:

To minimize the surface area of the box, we need to find the dimensions that satisfy the given conditions.

Let's let the side length of the square base be x and the height of the box be h. The volume of the box is given as 184 in³, so we have the equation x²h = 184.

To minimize the surface area, we need to minimize the sum of the areas of the square base and the four sides of the box. The surface area, A, is given by A = x² + 4xh.

To find the minimum surface area, we can solve for h in terms of x using the volume equation, substitute it into the surface area equation, and then find the value of x that minimizes the resulting expression using calculus.

The minimum surface area is √(552x² - 736x + 28864)