An Optimal Box PROJECT Imagine that you have to build a box with an open top out of a rectangular cardboard by cutting the four corners in equal parts and folding as depicted below. At the store, a rectangular cardboard costs 10 cents per square foot. The purpose of this project is to determine the box with maximum volume you can construct with the above procedure, if you have 90 cents to spend. Cost -wh = 90 X Volum = FOLD F F O : VL w,h,x ) L ! W-2 X h- 2x it to 1-we venable ox FOLD Whois = W- X2 Infolke h- x 2 2 = 2 Your solution should address all the following points (use complete sentences to explain your reasoning for each item below) V = (w- 2x ) (h- 2= ) x . Define all the variables you use and clearly describe what they represent. Also label the variables in a geometrical sketch [0.5 pt] . Translate the information into mathematical terms, using the variables you chose [ 0.5 pt] . Describe possible physical restrictions on the variables [0.5 pt] . Using an appropriate method, solve the optimization problem you obtained showing all steps and computations (no calculators allowed. Only methods discussed in class) [5 pt] . Among the solution/solutions you found, explain which one/ones gives/ give the maximum volume and why (if you use a test, show all computations). [1.5 pt] . What is the maximum volume you obtain? What are the dimensions of the cardboard that correspond to the box with maximum volume? What is the size of the cut that maximizes the volume? How much money is wasted on the portion of the cardboard that has to be cut off? [1.5 pt] . Can you give a geometric interpretation of the solutions you discarded? Use sketches and sentences to explain what happens in these cases [0.5 pt]