Answer the question below: A gift from an organization is wrapped in a box provided that the sum of length, width and height of the box do not exceed 96 inches. The student should find the dimensions of the box of maximum volume using "Lagrange multipliers" that will meet this condition.

I will accept a package whose combined height and perimeter of the base are at most 140 inches. Use Lagrange multipliers to find the dimensions of the largest volume box that will be accepted.

Find a rectangular piece of paper, cardboard, poster-board, etc. to create a box. Measure the sides of your paper and record them below (if you are using a normal size piece of paper its dimensions are 8.5 inches by 11 inches). You will now have to determine the length, width, and height to make an open-top box with the maximum volume possible. To do this you will have to cut the corners of the paper to fold up (but don't do this yet). Assume the height of the box will be x, which means you will need to subtract x from both sides of the width and length. Use these three dimensions to create a volume equation which you can derive like we did yesterday to find the optimal value for x leading to the maximum dimensions for your box. All of your work and the box you create should be submitted below.

A playpen is to be set up in a family’s recreation room using a 48 foot long flexible fence. Find thedimensions that will yield the maximum area if the playpen is built:a. In a rectangular shape in the middle of the room.b. In a rectangular shape using an existing wall for one side.