4. [-/9 Points] DETAILS A manufacturer wants to design an open-top box having a square base and a surface area of 144 square inches. What dimensions will provide a box with maximum volume? Length of base : inches Width of base : inches Height of box : inches Give all of your answers correct to two decimal places.[-/7 Points] DETAILS MY NOTES In this question, you will work step-by-step through an optimization problem. A craftswoman wants to make a cylindrical jewelry box that has volume, V, equal to 60 cubic inches. She will make the base and side of the box out of a metal that costs 20 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 craftswoman 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 ) U This is the objective function for this problem. b. Since the volume of the box is 60 cubic inches, complete the following constraint equation. 60 = Using the constraint equation, rewrite h in terms of r to 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. Ar 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 help6. [-/10 Points] DETAILS MY NOTES This question makes an important point: the maximum point of a function may not always be found by solving f '(x) = 0. . Remember: functions can have their minimum or maximum at an endpoint of their domain, at a point of non-differentiability (think of the absolute value function, which has its minimum point at zero) or may not even have a maximum or minimum. This means that the most thorough way of solving optimization problems involves sketching the objective function. (For questions that appear on tests, however, the optimum will usually occur at a relative max. or relative min. that can be found by solving f' ( x ) = 0 . ) Two parts of this question are multiple choice: you only get one submission attempt for those two parts. A peach orchard owner wants to maximize the amount of peaches produced by her orchard. She has found that the per-tree yield is equal to 1000 whenever she plants 35 or fewer trees per acre, and that when more than 35 trees are planted per acre, the per-tree yield decreases by 35 peaches per tree for every extra tree planted. For example, if there were 30 trees planted per acre, each tree would produce 1000 peaches. If there were 40 trees planted per acre, each tree would produce 1000 - 35 * (40 - 35) peaches, which is roughly equal to 825 peaches. a. Find the function that describes the per-tree yield, Y, in terms of x. Y= if x is no more than 35 trees per acre Y = if x is greater than 35 trees per acre b. Find the total yield per acre, T, that results from planting x trees per acre. T= if x is no more than 35 trees per acre T = if x is greater than 35 trees per acre c . Differentiate T with respect to x dT / dx = if x is less than 35 trees per acre dT / dx = if x is greater than 35 trees per acre Does this derivative ever equal zero? No Yes d. Sketch the graph of T as x varies and hence find the value of x that maximizes the yield and the maximum value of the yield. Optimal value of X : trees per acre Maximum yield : peaches per acre Is T differentiable when x equals 35? Yes No symbolic formatting help3. [-/9 Points] DETAILS MY NOTES 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 15 dollars per foot to construct. The front of the pen will be made of a special reinforced glass, which costs 60 dollars per foot to construct. The animal requires an area of 50 square feet in order to be comfortable. wood glass Figure 1. Schematic drawing of animal enclosure a. If the pen is to be built as cheaply as possible, what are the dimensions of the pen that should be built? Length of glass front of pen : feet Depth of pen : feet b. How much will the pen cost to build? Cost of pen : dollars Give all your answers correct to two decimal places.5. [-/7 Points] DETAILS A peach orchard owner wants to maximize the amount of peaches produced by her orchard. She cannot simply plant as many trees as she can, since planting more trees will decrease the amount of fruit that each tree produces (the yield of each tree). She has found that the per-tree yield can be described by the equation Y = 850 - 10 x. Here Y is the yield per tree and x is the number of trees planted per acre. For example, if there were 10 trees planted per acre, each tree would produce 850 - 10 * 10 = 750 peaches. Find the number of trees per acre that should be planted in order to produce the maximum crop and the resulting total yield. Number of trees per acre : trees per acre Total yield : peaches per acre Give your first answer correct to two decimal places, and the second answer correct to the nearest peach per acre