Math 116 Practice for Exam 2

Transcription

1 Math 6 Practice for Exam 2 Generated October 29, 205 Name: SOLUTIONS Instructor: Section Number:. This exam has 7 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 2. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the exam. 3. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam. 4. Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 5. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card. 6. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 7. You must use the methods learned in this course to solve all problems. Semester Exam Problem Name Points Score Winter interest 2 Fall potion 6 Fall Fall fishtank 5 Fall robot uprising 2 Winter coffee 8 Fall air filter 3 Total 9 Recommended time (based on points): 84 minutes

2 Math 6 / Exam 2 (March 23, 200) page 5 4. [2 points] A bank account earns 2.5% annual interest compounded continuously. Continuous payments are made out of the account at a rate of $5,000 per year for 8 years. a. [4 points] Write a differential equation describing the balance B = f(t), where t is in years satisfying 0 t 8. db dt =.025B 5,000 =.025(B 600,000) b. [4 points] Solve the differential equation you found in part (a) given an initial balance of B 0. db dt db (B 600, 000) ln B 600,000 =.025(B 600, 000) =.025dt =.025t + C B 600,000 = Ae.025t B = Ae.025t + 600,000 Given B = B 0 when t = 0, we have B 0 = A + 600,000, so A = B 0 600,000, giving B = (B 0 600,000)e.025t + 600,000. c. [4 points] What was the initial balance if the account has $0, 000 remaining 8 years after the account was opened? Give your answer to the nearest penny. Solving for B 0 given that B = 0,000 when t = 8, we have 0, 000 = (B 0 600,000)e.025(8) + 600,000, giving 590,000 = (B 0 600,000)e.45, which leads to B 0 = 590, 000e , , The initial balance would be approximately $223, Winter, 200 Math 6 Exam 2 Problem 4 (interest) Solution

3 Math 6 / Exam 2 (November 2, 204) page 4 3. [6 points] a. [5 points] Kazilla uses the ingredients you acquired from the store to make a special potion. Kazilla starts with a bucket containing 5 L of water and steadily adds purple liquid to the bucket at a rate of 0.5 L/min. Once in the bucket, the purple liquid mixes instantly with the water and the mixture drains out of the bottom of the bucket at a rate of 0.5 L/min. Let P(t) give the amount of purple liquid in the bucket t minutes after Kazilla starts making the potion. Write a differential equation involving P(t). dp dt = 0.5 P 0 b. [7 points] Later, Kazilla asks you to find the correct amount of green liquid to add to her potion. Let G(t) be the total amount of green liquid you need to add, in liters, after t minutes. Suppose G() = 3 2 and G(t) satisfies the differential equation dg dt = 2t(G )2. Find G(t). Start with a separation of variables: dg (G ) 2 = 2tdt (G ) = t2 +C G = t 2 +C + Using the initial condition that G() = 3/2, we can solve for C: 3 2 = +C + So G(t) = t C = Fall, 204 Math 6 Exam 2 Problem 3 (potion) Solution

4 Math 6 / Exam 2 (November 2, 204) page 5 c. [4 points] Finally, Kazilla asks you to add an amount of blue liquid B(t) at a rate given by db dt = 2B +et/2. For which value(s) of c is the function a solution to this differential equation? So 2c 5 = c 0 B(t) = 7e 2t + cet/2 5 B (t) = 4e 2t + cet/2 0 B (t) = db dt = 2(7e 2t + cet/2 5 )+et/2 = 4e 2t +( 2c 5 )et/2 or c = 2. Fall, 204 Math 6 Exam 2 Problem 3 (potion) Solution

5 Math 6 / Exam 2 (November 7, 200) page 5 4. [5 points] a. [5 points] Find the regions in the slope field of y = (y 3)y where the slopes are positive, negative or zero. Show all your computations. Positive : y > 3, y < 0 Zero: y = 0,3 Negative: 0 < y < 3 b. [2 points] Which of the following is the slope field of y = (y 3)y? Circle your answer (a) (b) B) c. [4 points] Find all the equilibrium solutions of y = (y 3)y. Use the slope field of the equation to determine the stability of each equilibrium solution. y = 3 unstable y = 0 stable d. [4 points] Let y(x) be the solution to the differential equation y = (y x)y satisfying y() = 2. Use Euler s method with steps x = 2 to approximate the value of y(2). Show all your computations. (x 0,y 0 ) = (,2) y = 2+(2 )(2)( 2 ) = 3 y 2 = 3+(3 3 2 )(3)( 2 ) = 5.25 then y(2) Fall, 200 Math 6 Exam 2 Problem 4 Solution

6 Math 6 / Exam 2 (November 7, 200) page 8 7. [5 points] An aquarium containing 00 liters of fresh water will be filled with a variety of small fish and aquatic plants. Awaterfilterisinstalledonthetanktohelpremovetheammoniaproducedbythedecomposing organic matter generated by plants and fish in the aquarium. The filter takes water from the tank at a rate of 20 liters every hour. The water then is filtered and returned to the aquarium at the same rate of 20 liters every hour. Ninety percent of the ammonia contained in the water that goes through the filter is removed. It is estimated that the fish and plants produce 30 mg of ammonia every hour. Assume the ammonia mixes instantly with the water in the aquarium. Fall, 200 Math 6 Exam 2 Problem 7 (fishtank) Solution

7 Math 6 / Exam 2 (November 7, 200) page 9 a. [6 points] Let Q(t) be the amount in mg of ammonia in the fish tank t hours after the fish were introduced into the aquarium. Find the differential equation satisfied by Q(t). Include its initial condition. ( ) ( ) dq Q Q dt = (.) dq = 30.8Q Q(0) = 0 dt b. [7 points] Find the amount of ammonia in the fish tank 3 hours after the fish were introduced into the aquarium. Make sure to include units in your answer. dq dt = 30.8Q dq 30.8Q = dt ln 30.8Q = t+c.8 Q(t) = De.8t = 66.6+De.8t using Q(0) = 0 you get Q(t) = 66.6( e.8t ) Q(3) = mg c. [2 points] What happens to the value of Q(t) in the long run? lim t Q(t) = lim t 66.6( e.8t ) = The value of Q(t) converges to 66.6 mg. Fall, 200 Math 6 Exam 2 Problem 7 (fishtank) Solution

8 Math 6 / Exam 2 (November 2, 204) page 2. [2 points] Franklin, your robot, is on the local news. Let R(t) be the number of robots, in millions, that have joined the robot uprising t minutes after the start of the broadcast. After watching the news for a little bit, you find that R(t) obeys the differential equation: dr dt = f(r) for some function f(r). A graph of f(r) is shown below. f(r) R a. [3 points] If R(t) is the solution to the above differential equation with R(0) = 0, what is R(t)? Justify your answer. lim t If R = 0, f(r) = R (t) is positive, so R will increase as t increases. As R increases to, R (t) = f(r) goes to 0, so lim t R(t) =. b. [6 points] Find the equilibrium solutions to the above differential equation and classify them as stable or unstable. R = 2 Stable Unstable R = Stable Unstable R = Stable Unstable R = 3 Stable Unstable c. [3 points] Let R(t) be a solution to the given differential equation, with R(3) = 0.5. Is the graph of R(t) concave up, concave down, or neither at the point (3,0.5)? Justify your answer. d 2 R dt 2 = d dt f(r) = f (R) dr dt = f (R)f(R) At R = 0.5, f (0.5) < 0 and f(0.5) > 0 so d2 R dt 2 < 0. Therefore, the solution curve will be concave down. Fall, 204 Math 6 Exam 2 Problem (robot uprising) Solution

9 Math 6 / Final (April 23, 200) page 7 5. [8 points] Kyle buys a nine-ounce cup of coffee at the store 0 minutes before class every morning. He doesn t drink the coffee until he arrives at class, ten minutes later. Suppose the coffee s temperature is initially 80 F when he buys it. a. [2 points] The temperature of the coffee in degrees Fahrenheit, T, at t minutes after it was purchased satisfies the differential equation dt dt = k(t 70), where k is a constant and 70 F is the surrounding air s temperature. Determine the units of k. Units of k are min. b. [4 points] Solve the differential equation given in part (a), finding T as a function of t. Your answer may include the constant k. dt = kdt T 70 dt = kdt T 70 ln T 70 = kt+c T 70 = Ae kt T = Ae kt +70 Since T = 80 when t = 0, we have T = 0e kt +70. c. [2 points] Suppose that Kyle adds one ounce of 40 F milk upon arriving at class, ten minutes after the coffee was purchased. When one ounce of milk is mixed with nine ounces of coffee, the resulting mixture has the temperature equal to [(Temp. of Milk)+9(Temp. of Black Coffee)]. 0 Determine the exact value of k if the temperature of the coffee-milk mixture is 00 F, immediately after the milk has been added. The temperature of the milk is 40 F and the temperature of the coffee 0 minutes after purchase is 0e 0k The temperature of the mixture is then 0 [40+9(0e0k +70)] = 0 ( e0k ), which equals 00 F. We solve for k: 0 ( e0k ) = e 0k = e 0k = 330 e 0k = 3 0k = ln(/3) ) k = 0 ln ( 3 Winter, 200 Math 6 Exam 3 Problem 5 (coffee) Solution

10 Math 6 / Exam 2 (November 6, 20) page 7. [3 points] A company designs an air filter for a ship s engine room that reduces the amount of fumes in the air by k percent every hour. The machinery in the engine room produces fumes at a rate of 0.02 kilograms per hour. Let Q(t) be the amount in kilograms of fumes in the room t hours after the engines are activated. Hence Q satisfies dq dt = 0.02 k 00 Q. a. [9 points] Find a formula for Q(t). Suppose there are no fumes in the air when the engines are activated. dq 0.02 k 00 Q = dt. 00 k ln 0.02 k 00 Q = t+c k 00 Q = Ae k 00 t Q(t) = 00 k Q(t) = 2 k (0.02 Ae k 00 t) ( e k 00 t) using Q(0) = 0. b. [2 points] What is the value of Q(t) in the long run? lim t Q(t) = 2 k. c. [2 points] Air safety regulations require that the concentration of fumes in the air not exceed 0 4 kilograms per liter at any time. If the volume of air in the engine room is 0 3 liters, for what values of k are the safety regulations met at all times? Concentration= Q(t) Volume 2 k Hence k 20. Fall, 20 Math 6 Exam 2 Problem 7 (air filter) Solution