[1 point) A box with no top is to be built by taking a sheet of cardboard measuring 13inches by 23inches, cutting zinch squares out of each comer, and folding up the sides. a} What are the length and width ofthe base of the box, after it is folded? It may help to draw a picture showing the cardboard before it is folded, and then draw another picture showing the res Length: 5 Width: ' b) What is the volume of the resulting box? Write your answer as a function of z. 'uI'OIU me: e) What is the domain ofthis function? In other words, what values of a: make sense? (Think about the original problem.) Domain: c!) Find all ofthe critical points of this function (notjust on the given domain). Critical points at: z = 3} Find the absolute maximum of this function within its domain. Absolute maximum: E cubic inches f) What size squares should be cut out of the comers to maximize the volume of the resulting bar? Size of squares: inches wide squares (1 point) A rectangular box has a volume of 25 cm" and has a square base. Find the dimensions of the box which minimize the surface area of the box Base: cm by cm Height: cm(1 point) A 2654 ft* tank (with a top) is to be built by welding four sides of height h to a square base of width a from sheet metal, and then a top the same size as the base is put on. The tank must also sit on a cement base 1 ft deep which should be the same size as the base of the tank. If the metal costs 22 dollars per square foot and the cement costs 10 dollars per cubic foot, what are the dimensions of the tank which minimizes the cost of the project? Width: Length: Height:(1 point) A hearing aid clinic is trying to determine how many batteries to purchase The more batteries the clinic purchases the cheaper the batteries are, However, storing the batteries costs money (in the form of renting space). The cost ofordenng 3 boxes of batteries is given by the function C(m] = 17:: 17 111(3). The cost of storing 2 boxes of batteries is given by the equation 3(3) : (108(2: 5UP, 0) Write an equation that models the total cost at purchasing and storing 2 boxes of batteries. Total Cost I?) Write an equation that models the cost per box of purchaSlng and storing 3 boxes of batteries. Cost per box: ::: dollars c] How many boxes of batteries should the clinic order in order to minimize the cost per box? Do NOT round your answer. HINT The equation 1717111 . - (\"+0.08 008 25007 0 z? 32 has solution x : 43.702. BOXES: EUAt what price should the clinic sell a box of batteries to ensure that the clinic earns a prot oi 23 dollars per bold Price per box: ::: dollars (\"i point) You are trying to visualize the graph 01 t) = t + 3111(t} on the interval [0, 1011']. Use optimization to determine what the "best" window is for viewing this graph. By "best." we mean the smallest window in which the entire graph is visible, so any smaller window would not show the entire graph, (Both of yourenlries in the answer blanks below should be intervals: the rst is the horizontal interval, and the second is the vertical interval determining the dimensions of the window.) 5851 Window: X (1 point) What are the dimensions of the closed cylindrical can that has surface area 441 cm- and contains the maximum volume? Radius: cm Height: cm