A rectangular solid with a square base has a volume of 5832 cubic inches. (Let x represent the length of the sides of the square base and let y represent the height.) (a) Determine the dimensions that yield the minimum surface area.
(b) Find the minimum surface area.

a) 18 in x 18 in x 18 in

b)

Step-by-step explanation:

a) Let's call 's' the side of the square base and 'h' the height of the solid.

The surface area is given by the equation:

The volume of the solid is given by the equation:

From the volume equation, we have that:

Then, using this value of h in the surface area equation, we have:

To find the side length that gives the minimum surface area, we can find where the derivative of S in relation to s is zero:

The height of the solid is:

b) The minimum surface area is:

both equations and proportions are better.