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Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions a, = 9 inches by b = 8 inches by cutting a square of side a: at each corner and turning up the sides (see the figure). Determine the maximum possible volume of this box. Following the steps to solve the problem. Check Show Answer only after you have tried hard. (1) Express the volume V as a function of cc: V = _ (2) Determine the domain of the function V of a: (in interval form): (3) Find the derivative of the function V: V' = _ (4) The value of a: that maximizes the volume is: in (5) The maximum volume is V = m3 A box with a square base and open top must have a volume of 108000 cm3. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only m, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of 93.] Simplify your formula as much as possible. AW Next, find the derivative, A '(m). w = Now, calculate when the derivative equals zero, that is, when A'(:c) = 0. [Hint: multiply both sides by m -] We next have to make sure that this value of :1: gives a minimum value for the surface area. Let's use the second derivative test. Find A".(a:) m = Evaluate A"(m) at the m-value you gave above. [:1 NOTE: Since your last answer is positive, this means that the graph of A(a:) is concave up around that value, so the zero of A ' (9;) must indicate a local minimum for 11(3). (Your boss is happy now.) 2 What is the area of the minimum amount of material used? [:sz Glorious Gadgets is a retailer of astronomy equipment. They purchase equipment from a supplier and then sell it to customers in their store. The function C(91) = 3m + 19125m' 1 + 12750 models their total inventory costs (in dollars) as a function of a: the lot size for each of their orders from the supplier. The inventory costs include such things as purchasing, processing, shipping, and storing the equipment. What lot size should Glorious Gadgets order to minimize their total inventory costs? (NOTE: your answer must be the whole number that corresponds to the lowest cost.) C] What is their minimum total inventory cost? SC] A cylinder shaped can needs to be constructed to hold 400 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.07 cents per square centimeter. Find the dimensions for the can that will minimize production cost. Helpful information: h : height of can, r : radius of can Volume of a cylinder: V = arrzh Area of the sides: A = 21r'rh Area of the top/ bottom: A = 11"!'2 To minimize the cost of the can: A poster has upper! lower margins of 3 inches and left/ right margins of 2 inches. The printed part of the poster(inner rectangle) is A1 = 96 square inches. The goal is to find the dimesnions of the (entire)poster that minimizes the (entire) area. The area of the entire poster, as a function of a: and y is Use the substitution principle to rewrite the area as a function of an. (Hint: A1 : a: - 3;) MW: To determine optimal values, we need to look at A'(a:) = 0. mm = :1 Solving for a: and, ultimately for 3;, yields :5 = C] inches 3; = C] inches Finally, the dimensions of the entire poster, that minimizes the area, are C] inches by C] inches Consider a rectangle inscribed in a right triangle with sides A : 7 and B : 21. We want to determine the dimensions of the rectangle with largest area that can be inscribed inside the triangle. A B .y x Il A The area of the rectangle, as a function of .1: and y is A=C1 Use the substitution principle to rewrite the area as a function of (1:. (Hint: Similar Triangles) A=A