3:56 M 9 9 1 61% mylab.pearson.com/Stu 16 ... MATH.IZZU-084 online spring ZUZ3 Sonal Kana 04/20/23 3:55 PM Question 20, 4.1.71-GI HW Score: 57%, 57 of 100 Homework: MLM #9 Part 1 of 6 points Points: 0 of 5 Save Question list View an example | All parts showing X 0.06t for Betaburgh, when 1, Alphaville had a Question 1 The rates of change in population for two cities are P'(t) = 47 for Alphaville, and Q'(t) = 1040.05t for Betaburgh, where t is the number of years since 1990, and P' and Q' are measured in people per year. In 1990, Alphaville had a population of 5200, and Betaburgh had a population of 3600. Answer parts a) through c). Question 2 Question 3 O Question 4 a) Determine the population models for both cities. The population model for each city is the antiderivative of the rate of change of its population. Find the population model by integrating, and then apply the given initial conditions to determine the value of the Question 5 constant of integration. Find the antiderivative of P'(t) = 47, the rate of change of Alphaville's population. P' (t) = 47 O Question 6 P(t) = 47 dt = 47t + C Question 7 Therefore, the antiderivative of P'(t) is 47t + C. To determine the constant of integration C, apply the given initial condition P(0) = 5200. Substitute 0 for t and then solve for C. Question 8 P(t) = 47t+ C 5200 = 47(0) + C Substitute. 5200 = C Simplify Question 9 The population model for Alphaville is P(t) = 47t + 5200. Next find the antiderivative of Q'(t) = 1040.5, the rate of change of Betaburgh's population. Question 10 Q'(t) = 1040.05t Q (t) = 104 0.05t at Question 11 Move the constant factor to the front of the integral, then integrate | 0.05t dt. Question 12 Q (t) = 104 0.05t at x Question 13 = 104 0.05t at = 2080e0.05t + c O Question 14 Therefore, the antiderivative of Q'(t) is 2080e + C. To determine the constant of integration C, apply the given initial condition Q(0) = 3600. Substitute 0 for t and then solve for C. (t) = 2080e .05t + C O Question 15 3600 = 2080e0.05(0) + c Substitute. 3600 = 2080 + C Simplify. Question 16 Solve for C. 3600 = 2080 + C 1520 = C Question 17 The population model for Betaburgh is Q(t) = 2080e -05 + 1520. Question 18 b) What were the populations of Alphaville and Betaburgh, to the nearest hundred, in 2000? To determine the population of each city, substitute the value of t corresponding to the indicated year into each population model found in part a). Note that the value of t that corresponds to the year 2000 is t = 10. Question 19 Substitute t = 10 into the population model for Alphaville, P(t) = 47t + 5200, and evaluate, rounding to the nearest hundred. P(t) = 47t+ 5200 Question 20 P(10) = 47(10) + 5200 ~ 5700 Substitute t = 10 into the population model for Betaburgh, Q(t) = 2080e+ 1520, and evaluate. Round to the nearest hundred. Q(t) = 2080e0.05t + 1520 Q(10) = 2080e0.05(10) + 1520 ~4900 Print Close Help me solve this View an example Get more help - Clear all Check answerIn the year 2000, the populations of Alphaville and Betaburgh were 5700 and 4900, respectively. c) Sketch the graph of each city's population model and estimate the year A Population (1,000s) in which the two cities have the same population. 10- Notice that the population model for Alphaville, P(t) = 47t + 5200, is a linear function. The graph of P(t) is shown at right. To graph the population model for Betaburgh, Q(t) = 2080e. + 1520, create a table of values, plot the resulting points, and draw a smooth curve through the plotted points. Find the corresponding value of Q(t) for each value of t. 0 10 15 20 25 Q(t) = 2080e0.05 + 1520 3600 4900 5900 7200 8800 Therefore (0,3600), (10,4900), (15,5900), (20,7200), and (25,8800) are points A Population (1,000s) on the graph of Q(t). They are shown plotted on the graph at right. 10- 25 Draw a smooth curve through the plotted points. A Population (1,000s) 10- Q Use the above graph to estimate the year in which two cities have the same population. The year is the one corresponding to the value of t at which the graphs intersect. The two cities have the same population in the year 2005. Print CloseTH.1220-082 online spring 2023 Sonal Rana 04/26/23 3:57 PM (? HW Score: 57%, 57 of 100 Homework: MLM #9 Question 20, 4.1.71-GI Part 1 of 6 points O Points: 0 of 5 Save estion list K The rates of change in population for two cities are P'(t) = 45 for Alphaville and Q'(t) = 105e.for Betaburgh, where t is the number of years since 1990, and P' and Q' are measured in people per year. In 1990, Alphaville had a population of 6000, and Betaburgh had a population of 3500. Answer parts a) through c) Question 1