(1 point) A box with no top is to be built by taking a sheet of cardboard measuring 13-inches by 23-inches, cutting -inch squares out of each corner, and folding up the sides. a) What are the length and width of the 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 resulting box. Length: Width: b) What is the volume of the resulting box? Write your answer as a function of I. Volume: c) What is the domain of this function? In other words, what values of a make sense? (Think about the original problem.) Domain: d) Find all of the critical points of this function (not just on the given domain). Critical points at: I = e) Find the absolute maximum of this function within its domain. Absolute maximum: cubic inches f) What size squares should be cut out of the corners to maximize the volume of the resulting box? Size of squares: -inches wide squares(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 of ordering a boxes of batteries is given by the function C(@) = 17x - 17 In(). The cost of storing a boxes of batteries is given by the equation S(x) = 0.08(x - 50)2 a) Write an equation that models the total cost of purchasing and storing a boxes of batteries. Total cost: dollars b) Write an equation that models the cost per box of purchasing and storing a 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 17 - 17 In(x) + 0.08 - 0.08 . 2500 0 has solution c = 43.702. Boxes: d) At what price should the clinic sell a box of batteries to ensure that the clinic earns a profit of 28 dollars per box? Price per box: dollars(1 point) You are trying to visualize the graph of f(t) = t + sin(t) on the interval |0, 10x]. 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 your entries in the answer blanks below should be intervals: the first is the horizontal interval, and the second is the vertical interval, determining the dimensions of the window.) Best Window: X(1 point) A 2654 fts 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) What are the dimensions of the closed cylindrical can that has surface area 441 cm* and contains the maximum volume? Radius: cm Height: cm