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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 20 dollars per foot to construct. The front of the pen will be made of a special reinforced glass, which costs 40 dollars per foot to construct. The animal requires an area of 40 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 : x feet Depth of pen : x feet b. How much will the pen cost to build? Cost of pen :x dollars Give all your answers correct to two decimal places.A peach orchard owner wants to maximize the amount of peaches produced by his orchard. He cannot simply plant as many trees as he can, since planting more trees will decrease the amount of fruit that each tree produces {the yield of each tree). He has Found that the per-tree yield can be described by the equation Y = 1000 - 20 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 1000 - 20 * 10 = 800 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 :\\:| x trees per acre Total yield : ':| x peaches per acre Give your rst answer correct to two decimal places, and the second answer correct to the nearest peach per acre. 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 1200 whenever she plants 70 or fewer trees per acre, and that when more than 70 trees are planted per acre, the per-tree yield decreases by 20 peaches per tree for every extra tree planted. For example, if there were 65 trees planted per acre, each tree would produce 1200 peaches. If there were 75 trees planted per acre, each tree would produce 1200 - 20 * (75 - 70) peaches, which is roughly equal to 1100 peaches. a. Find the function that describes the per-tree yield, Y, in terms of x. Y = if x is no more than 70 trees per acre Y = if x is greater than 70 trees per acre b. Find the total yield per acre, 7, that results from planting x trees per acre. T = if x is no more than 70 trees per acre T= if x is greater than 70 trees per acre c. Differentiate T with respect to x dT / dx = if x is less than 70 trees per acre dT / dx = if x is greater than 70 trees per acre Does this derivative ever equal zero? O No O Yes d. Sketch the graph of 7 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 7 differentiable when x equals 70? O No O Yes
