Answer Happy

(36 points) Area₁ 80 70 60 50 40 ✔B Area2 C Area3101.9 -2 30 20 10 -1 0 -10 -20 n=2 10 11 C'(x) S'(x) 12 13 (2
Puddy's Auto Shop recently purchased a new hydraulic lift, engine crane, and electronic diagnostic equipment. When x represents the number of years the equipment is in use, then the new equipment has an associated rate of savings given by S'(x) = 52.668-0.2688x² tens of dollars per year. The equipment also has an operating expense with an associated rate of cost given by C'(x) = 0.2688x² + 1.0136x tens of dollars per year, when the equipment is in use for x years. The applet above shows the graph of S'(x) in blue and C'(x) in red. Checkboxes A, B, and C will display three different regions with their respective areas shaded related to this problem. The slider for n will allow you to control the upper limit for these regions. Use the applet to answer the questions below. (a) Determine how long the new equipment should be used in order to maximize savings and use your answer to complete the sentences below. (Hint: compare S' to C' before and after point I in the graph). In order to maximize their savings, Puddy's Auto Shop should use the new equipment for a total of years. At this time, the rate of savings and the rate of cost will both be equal to and cost are both equal to $ tens of dollars per year (round to 1 decimal place). Equivalently, the rates of savings per year (round to the nearest whole dollar). (b) Set up and evaluate the definite integral to determine the accumulated cost of using the new equipment during the time period determined in part (a). For the last blank, select the corresponding area given by the definite integral (the area in the graph that represents accumulated costs). Accumulated Costs = dx (round to 1 decimal place)
(c) Set up and evaluate the definite integral to determine the accumulated savings of using the new equipment during the time period determined in part (a). For the last blank, select the corresponding area given by the definite integral (the area in the graph that represents accumulated savings). Accumulated Savings = dx (round to 1 decimal place). Area 1 (d) Set up and evaluate the definite integral to determine the NET accumulated savings of using the new equipment during the time period determined in pait (a). Notice that this answer will also be the maximal possible savings from the new equipment (again, using the answer from (a) above). For the last blank, select the corresponding area given by the definite integral (the area in the graph that represents the net accumulated savings). Net Accumulated Savings = dx (round to 1 decimal place) Area 3 Note: This answer should be equal to the difference between the answers to parts (b) and (c), but could be slightly off due to rounding error. However, the
Note: This answer should be equal to the difference between the answers to parts (b) and (c), but could be slightly off due to rounding error. However, the answer should match the value from the applet. (e) Complete the following to interpret the answer from part (d). Also, to verify that the answer to (d) does indeed give the maximal savings, compare it to the savings if the equipment is used for more or less time. Convert all net savings amounts to dollars. (Hint: Use the applet to compute the requested comparisons). years. In this time, Puddy's total net Puddy's Auto Shop will maximize their savings when they use the new equipment for a span of savings will be $ If the shop decides to keep using the equipment for one additional year-for a total of net savings will instead be a total of $ Similarly, if the equipment is discontinued one year early and only used for then the net savings for Puddy's Auto Shop will be $ years- then the years.