Exponential models #5 - Questions and Answers

Question 1 The following table shows the tuition for a semester at CBU in dollars for years since 1990. Year | CBU Tuition 2 | 3995 5 | 5000 8 | 6200 11 | 7450 14 | 8855 17 | 10420 20 | 12160 A scatter plot for the data and a graph of a linear model in black and a quadratic model in red follows. 3691215182140005000600070008000900010000110001200013000 Use the above scatter plot to decide which model better fits the data. none quadratic linear Let C(t) be the cost of tuition at CBU in dollars for t years since 1990. A quadratic model for the data is c(t) = Use three decimals in your answer. Estimate the tuition in 2016. $ Use the model to predict the year in which the tuition will be $15325. Answer x=number of years; y= CBU tuition. as we can see from scatter plot, here the best-fit model if the quadratic model, so answer is quadratic. Use calculator to find the quad- ratic regression. So, equation is y = ax² + bx + c y = 7.994x² + 276.033x + 3420.364 C(t)=7.994t² + 276.033t + 3420.364 (t=27) C(27)=7.994 (27)² + 276.033(27) + 3420.364 C(27)=5827.626 + 7452.891 + 3420.364 C(27)=16700.881 ≈ 16701 - Estimate the tuition in 2017 Note: you may need to round answer per the given instruction C = 15550 7.994t² + 276.033t + 3420.364 = 15550 7.994t² + 276.033t + 3420.364 - 15550 = 0 7.994t² + 276.033t - 12129.636 = 0 t² + 34.53t - 1517.343 = 0 t = -b ± √(b² - 4ac)/2a t₁₂ = -34.53 ± √(-34.53)² + 4 * 1 * 1517.343 / 2 * 1 t₁₂ = -34.53 ± √1192.3209 + 6069.372 / 2 t₁₂ = -34.53 ± √7261.6929 / 2 t₁₂ = -34.53 ± 85.21556 / 2 take positive value only t = (-34.53 + 85.21556) / 2 t = 25.342 t ≈ 25 years after 1990 the tuition will be $15550 in 2015 Question 2 In millions, t years after 2010, the exponential models describe the population of the indicated country, A. Which country has the greatest growth rate? How much does that nation's population grow annually, on average? India, A = 1173.1e⁰⁰⁰⁸ᵗ Iraq, A = 31.5e⁰⁰¹⁹ᵗ Japan, A = 127.3e⁰⁰⁰⁶ᵗ Russia, A = 141.9e⁻⁰⁰⁰⁵ᵗ Answer The exponential model that describes the population of the indicated country A, in millions,t years after 2010 is India, A = 1173.1e⁰⁰⁰⁸ᵗ Iraq, A = 31.5e⁰⁰¹⁹ᵗ Japan, A = 127.3e⁰⁰⁰⁶ᵗ Russia, A = 141.9e⁻⁰⁰⁰⁵ᵗ Here value of k represents the growth rate of population. Comparing the exponential model from Standard equation A = A₀eᵏᵗ, value of k for country India is 0.008, for Iraq is 0.019, for Japan is -0.006 and for Russia is -0.005 Since country Russia and Japan has negative value of k, Rightarrow the decay rate or population is decreasing as the time increases. Now arrange the positive value of in descending order: Since the country Iraq has greatest value of k, so the country Iraqgreatest growth rate. The growth rate of population of Iraq each year is 0.019 * 100 = 1.9% Question 3 The polynomial -16t² + vt + ho models the height, in feet, of a ball t seconds after it is thrown. In this polynomial, v represents the initial vertical velocity of the ball and ho represents the initial height of the ball. Min throws a baseball from a height of 6 feet with an initial vertical velocity of 30 feet per second. Which polynomial models the height of the ball, in feet? Answer According to the question, the ball has a starting height of 6 feet and a vertical velocity of 30 feet per second (v). So substituting these into the model polynomial, we get -16t² + 30t + 6 Question 4 A researcher is trying to determine the doubling time of a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. Use a graphing calculator to find an exponential curve f(t)=abᵗ that models the bacteria population t hours later. Time (hours) 0 4 8 12 16 20 24 Bacteria count (CFU/mL) 37 47 63 78 105 130 173 Answer Using the given data in a graphing calculator, we determine an exponential curve f(x) = abˣ that models the data. The function is f(t) = 36.7826 * 1.06633ᵗ yl ~ a*bˣ R² = 0.9986 el plot a = 36.7826 b = 1.06633 29.2s