Among the specifications of a company, which requires packing its product in boxes, it is found that for a box the sum of its dimensions (length (x) -width (y) -height (z)) must be 450 cm. The function that allows determining the maximum volume of the box that meets these characteristics is given by:

A cylindrical can with no top is to be constructed for 20 pesos. The cost of the material used for the bottom is 2 pesos per square inch and the cost of the material used for the curved side is 3 pesos per square inch A. Express the volume of the can as a function of its radius B. What are the dimensions that yield a maximum volume

A box with a square base and with height f(x) is to be constructed so that it has a volume of 1000 cubic centimeters. What are the possible dimensions of the box?

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.