This question has several parts that must be completed sequentially. if you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. tutorial exercise a rectangular box which is open at the top can be made from an 24-by-30-inch piece of metal by cutting a square from each corner and bending up the sides. find the dimensions of the box with greatest volume, where h = height, l = length, and w = width. (note: let the width be determined by the 24-inch side and the length by the 30-inch side.)

The maximum volume is 1417.87

Explanation:

Optimization Using Derivatives

We have a 24x30 inch piece of metal and we need to make a rectangular box by cutting a square from each corner of the piece and bending up the sides. The width of the piece is 24 inches and its length is 30 inches

When we cut a square of each corner of side x, the base of the box (after bending up the sides) will be (24-2x) and (30-2x), width and length respectively. The volume of the box is

Operating

To find the maximum value of V, we compute the first derivative and equate it to zero

Simplifying by 12

Completing squares

We have two values for x

The first value is not feasible because it will produce a negative width (24-2(13.58))=-6.16

We'll keep only the solution

The width is

The length is

And the height

The maximum volume is

if they have a finite resultant