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O webassign.net C U Studio, 1, 2, 3, 4 & 5 Bedroom Student Apartments in Raleigh, NC VA Optimization - MA 131, section 002, Fall 2020 | WebAssign height : inches g. What is the maximum volume of the box? volume : cubic inches symbolic formatting help 2. [-/7 Points] DETAILS MY NOTES In this question, you will work step-by-step through an optimization problem. A craftsman wants to make a cylindrical jewelry box that has volume, V, equal to 45 cubic inches. He will make the base and side of the box out of a metal that costs 50 cents per square inch. The lid of the box will be made from a metal with a more ornate finish which costs 100 cents per square inch. a. Writing the radius of the cylindrical box as r, and the height of the box as h, calculate the cost, C, in cents, of the metal used to produce the box in terms of h and r. (Assume that the craftsman only need consider the metal that actually ends up in the box: no metal is wasted.) (WebAssign note: remember that you enter 7 as pi ) C = This is the objective function for this problem. b. Since the volume of the box is 45 cubic inches, complete the following constraint equation. 45 = Using the constraint equation, rewrite h in terms of r h = c. Rewrite your expression for the cost of the box in terms of the single variable r. C = d. Differentiate C with respect to r, to find the derivative dC / dr. do dr = e. Find the value of r for which we have a potential relative extreme point of C. (Give your answer, and those that follow, correct to two decimal places.) r = For this question, you need not show that this is actually a relative minimum point. (But you should know how you would do this.) f. What is the height of the box? h = symbolic formatting help 3. [-/9 Points] DETAILS A prize winning animal is going to be shown at the State Fair. A pen must be constructed to hold the animal. The rectangular pen will have three wooden sides, which cost 25 dollars per foot to construct. The front of the pen will be made of a special reinforced glass, which costs 45 dollars per foot to construct. The animal requires an area of 70 square feet in order to be comfortable