370 CHAPTER 5 Exponential and Logarithmic Functions (a) See Figure 69 for a scatter plot of the data on a TI-84 Plus CE. (b) A graphing utility fits the data in Table 11 to a logistic growth model of the form = + − y c ae 1 bx by using the LOGISTIC regression option. Figure 70 shows the result on a TI-84 Plus CE. The logistic model from the data is = + − y e 663.0 1 71.6 x 0.5470 where y is the amount of yeast biomass in the culture and x is the time. (c) See Figure 71 for the graph of the logistic model on a TI-84 Plus CE. Figure 72 shows the logistic model using Desmos. Now Work PROBLEM 7 Applications and Extensions 5.9 Assess Your Understanding 1. Covid-19 In 2020, the world was exposed to a novel coronavirus called Covid-19.The virus resulted in a pandemic throughout many countries. In the United States, the number of individuals infected with Covid-19 grew rapidly in the early stages of the disease reaching the country. The data below represent the cumulative number of documented cases of Covid-19 in 2020 during the early phase of the disease. Let x represent the number of days since February 20, 2020. Date Cumulative Number of Cases Feb 20 ( ) = x 0 15 Feb 25 ( ) = x 5 57 March 1 ( ) = x 10 75 March 6 ( ) = x 15 319 March 11 ( ) = x 20 1301 March 16 ( ) = x 25 4596 March 21 ( ) = x 30 24,192 March 26 ( ) = x 35 85,435 March 31 ( ) = x 40 188,530 Source: https://www.worldometers.info/coronavirus/country/us (a) Draw a scatter plot treating days since February 20 as the independent variable. (b) Using a graphing utility, build an exponential model from the data. (c) Express the function found in part (b) in the form ( ) = N t N e . kt 0 (d) Graph the exponential function found in part (b) or (c) on the scatter plot. (e) Use the exponential function from part (b) or (c) to predict the cumulative number of cases of Covid-19 on April 5 ( ) = x 45 . (f) Use the exponential function from part (b) or (c) to predict when the cumulative number of cases reached 1,000,000. Figure 69 TI-84 Plus CE 0 22 700 20 Figure 70 Logistic model using a TI-84 Plus CE Figure 71 0 22 700 20 Solution Figure 72 Logistic model using Demos (d) Based on the logistic growth model found in part (b), the carrying capacity of the culture is 663. (e) Using the logistic growth model found in part (b), the predicted amount of yeast biomass at = t 19 hours is = + ≈ − ⋅ y e 663.0 1 71.6 661.5 0.5470 19 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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