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Centerville is the headquarters of Greedy Cablevision inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation. Centerville is located at [11'], D} in the mgr-plane, Springeld is at {0, 'i'}. and Shelbyville is at (D, 7}. The cable runs from Centerville to some point {3, D} on the zaxis where it splits into two branches going to Springeld and Shelbyville. Find the location (as, [i] that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer. To solve this problem we need to minimize the following function of a: rim) =D We nd that x} has a critical number at m 2E] To ven'fy that re} has a minimum at this critical number we compute the second derivative f"{.r] and nd that its value at the critical number is D a positive number. Thus the minimum length of cable needed is m A rancher wants to fence in an area of 1000000 square feet in a rectangular field and then divide it in half with a fence down the middle, parallel to one side. What is the shortest length offence that the rancher can use? Length offence = feet. An open box is to be made out of a 8-inch by 20-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume. w in X Y L in L - 2x in w - 2x in Dimensions of the bottom of the box: X Height of the box:Find the dimensions of the rectangle with area 169 square inches that has minimum perimeter, and then nd the minimum perimeter. 1. Dimensions: 2. Minimum perimeter: Enter your result for the dimensions as a comma separated list of two numbers. Do not include the units. A pig farmer wants to enclose a rectangular area and then divide it into six pens with fencing parallel to one side of the rectangle (see the figure below}. There are 900 feet of fencing available to complete thejob. What is the largest possible total area of the six pens? [D333] Largest area = I I {include _!__I_r_1_i_t_5_._] If 1300 square centimeters of material is available to make a box with a square base and an open top. nd the largest possible volume of the box. Note that the surface area of the box is given by 43;; + 33 as depicted in the image. 1' X "volume = C] (include units]