Exam IIIC - Math 1743 Fall 2007 Exam HI FORM C Name Key...

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Unformatted text preview: Math 1743 Fall 2007 Exam HI FORM C Name: Key Section #: i.d.#: Instructor: Read all questions carefully and answer them completely. Show appropriate work to receive maximum credit. In general, if you can label something with units, you should label it with units. When computing and writing numerical answers, use rounding guidelines discussed in class unless the problem specifies otherwise. Every model you write (and that includes rate of change models) should be complete. You will have 75 minutes for the exam and point values are as indicated. 1. Suppose that the population of a certain collection of Brazilian ants is given by A(t) = (t + 300)- In (t + 4) ants after t days. (15 points) a. Find the complete rate of change model for the population of the ants, A'(t). A(t) = (1).1n(t +4)+ (t +300). “+4 :1n(t+4)+ trio ants per day, afier t days. b. What was the ant population after 20 days? A(20) = 1016.98; 1016 or 1017 ants c. Find and interpret A’(20). A’(20)= 16.5; After twenty days, the collection of ants was growing by 16.5 ants per day. 0n problems 2—4, you do n_ot need to simplzfi/ your answers. 2. Given g(x) = 79K + (9 — x)7, find g’(x). (8 points) g’(x) = (1n 7)79x 9+ 7(9—x)6 -—1 3. Given S(p) = 65.823p(0.904p), find S'(p). (8 points) S'(p) = 65.823~1(0.904p)+65.823p~1n 0.904(0.904P) 4. Given v(x) = xikfl find Q (11 points) e -(In x) dx v(x) = 37e‘x - (In x)—6 V'(x) = 37e‘x -—1 - (In x)"6 + 37e’” -—6- (In x)‘7 i X 01‘ v(x) = 37(ex - (1n X)6 )—1 V'(x) = 37- —1(ex -(1n x)6 )_2 [ex - (In x)6 + ex -6(ln x)5 -l) 5. The weekly weight loss of a person on a weight-loss program is given by w(t) = 1.58te‘°'lgt pounds after t weeks on the program. (17 points) In parts a & b, give both input and output, round coordinates to the hundredths place, and please label them with units. a. Find the absolute maximum weekly weight loss between weeks three and eight on the program. t = 5.56 weeks w = 3.23 pounds b. Find the absolute minimum weekly weight loss between weeks three and eight on the program. t = 3 weeks w = £2.76 pounds 0. Based on your answers to the questions above, fill in the following: On the interval [3, 8], w’(t) is positive between t = 3 and t = 5.56 . On the interval [3, 8], w'(t) is negative between t = 5.56 and t = 8 . 6. A soda bottling company bottles 15,000 cases of orange soda each year. The cost to set up the machine for a production run is $1300, and the cost to store a case for one year is $12. (19 points) a. If x is the number of cases in production run, write an expression representing the number of production runs necessary to produce 15,000 cases. 15000 x b. Write an expression for the total set up cost. [15000](1300) X c. Assume that the average number of cases stored throughout the year is half the number of cases in production run. Write an expression for the total storage cost for one year, using x as the number of cases in each production run. a» (1. Use your responses to parts b & c to write a complete model for the combined set up and storage costs for one year. C(X) = (15000](1300)+[%)(12) = W + 6x dollars, where x cases are in each x x production run. e. What size production run minimizes the total yearly cost? Show some work (i.e. calculus) to support your answer. Round to the nearest whole number. C(x) =19500000x—1 + 6x ; C’(x) = —19500000x—2 + 6 =0 x = 1802.78; 1802 or 1803 cases f. How many production runs will minimize total yearly production cost? Round to the nearest tenth, assuming they can make a partial run to end up with 15,000 cases. 8.3 runs 7. An international wildlife agency determined that the population of Ozark big-cared bat for given years is: (22 points) 1972 . 1977 1982 1987 1992 1997 2002 Ozark big—eared bats 87.5 81.4 63.2 39.8 22.7 18.9 17.0 (thousand bats) a. Find a complete logistic model for this data. Align the data so that the year 1972 corresponds to an input of zero. 114.257 f x 2—— ( ) 1+O.26le°'l23x thousand bats, x years after 1972 =114.257(1+ 026161123")—1 b. Find the complete model for rate of change of the number of population of Ozark big- eared bats. f’(x) = 114.257 - —1(1+ 0.261e0'123" )‘2 -0.261e°“23" - 0.123 = —3.668e°‘123x (1 + 0.261e0'123x )_2 thousand bats per year, x years afier 1972 c. Find the inflection point for the model you generated in part a. Round both coordinates to the hundredths place and label them with appropriate units. x = 10.96 years after 1972; f = 57.12 thousand bats ...
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