Please help me with this problem: 1) A box with a square base and an open top must have a volume of 32,000cm3. Your goal is to find the dimensions of the box that minimize the amount of material used. Show all your work. Label you work in an orderly way. Put your answers in a box. a) Draw a picture and label all variables that you will use. b) Write down the equations for the volume and surface area using those variables. c) Calculate the desired dimensions. d) Show that your answer really gives a minimum by using either the first or second derivative test.

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.

Isla wants to build a playhouse that is in the shape of a rectangular pyramid that has the maximum volume possible. The perimeter of the base must be 24 feet. The height of the playhouse should be 2 times the length. What should the dimensions of the play house be? a. 3 feet wide by 9 feet long by 18 feet high b. 6 feet wide by 6 feet long by 12 feet high c. 5 feet wide by 7 feet long by 14 feet high d. 4 feet wide by 8 feet long by 16 feet high