Show all work on a separate sheet of paper with justifications where necessary. Answers should be clearly marked or boxed. Print first and last names of each group member at the top of the page. 1. Formulating an objective function. (a) According to postal regulations, a shipping box is classified as "oversized" if the sum of its height and girth (perimeter of its base) exceeds 108 in. Write an objective function of a single variable that gives the volume of a box with square base that is not oversized. Be sure to give a reasonable domain. (b) A landscape architect wishes to enclose a rectangular garden of area 1000 m on one side by a brick wall and by metal fencing along the other three sides. The cost for the brick wall is $90/m and the cost for metal fencing is $30/m. Write an objective function of a single variable that gives the cost of the project. Be sure to give a reasonable domain. (c) A poster of area 6000 cm has blank margins of width 10 cm on the top and bottom and 6 cm on the sides. Write an objective function of a single variable for the area of the printed region inside the margins. Be sure to give a reasonable domain. 2. Find two positive numbers whose product is 100 and whose sum is a minimum. 3. Minimizing Cost A builder wants to construct a cylindrical barrel with a capacity of 327 cubic feet. The cost per square foot of the material for the side of the barrel is half that of the cost per square foot for the top and bottom. Determine the dimensions of the barrel that can be constructed at a minimum cost in terms of material used. 4. Minimizing Distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)? 5. Minimizing Perimeter - Of all rectangles with a fixed area of A inches squared, which one has the smallest perimeter P? (a) In an attempt to solve this problem, Antonio produced the following preliminary work. Review his work shown below. If Antonio approached you for advice on setting this problem up, how would you respond? Constraint: Objective Function: Substitute the constraint, W = - - L Perimeter = 2(Length) + 2(Width) Area = (Length) . (Width) A = L. (7 - L) P = 21 + 2W A = L . W A = =. L - 12 Solving for W we have, P - 2L P W = - L Reasonable domain: 0 S LS - 2 2 (b) Show your own work to solve this problem. Give the dimensions in terms of A