SECTION 5.9 Building Exponential, Logarithmic, and Logistic Models from Data 369 Now Work PROBLEM 5 3 Build a Logistic Model from Data Logistic growth models can be used to model situations for which the value of the dependent variable is limited. Many real-world situations conform to this scenario. For example, the population of the human race is limited by the availability of natural resources, such as food and shelter. When the value of the dependent variable is limited, a logistic growth model is often appropriate. Figure 67 20.2 525 2.4 775 Figure 68 Logarithmic model using Desmos Figure 66 Logarithmic model using a TI-84 Plus CE (c) Figure 67 shows the graph of ( ) = − h p p 45.7863 6.9025 ln on the scatter plot. Figure 68 shows the logarithmic model using Desmos. (d) Using the function found in part (b), Jodi predicts the height of the weather balloon when the atmospheric pressure is 560 to be ( ) = − ≈ h 560 45.7863 6.9025 ln 560 2.108 kilometers Fitting a Logistic Function to Data The data in Table 11 represent the amount of yeast biomass in a culture after t hours. EXAMPLE 3 Time (hours) Yeast Biomass Time (hours) Yeast Biomass Time (hours) Yeast Biomass 0 9.6 7 257.3 14 640.8 1 18.3 8 350.7 15 651.1 2 29.0 9 441.0 16 655.9 3 47.2 10 513.3 17 659.6 4 71.1 11 559.7 18 661.8 5 119.1 12 594.8 6 174.6 13 629.4 Source: Tor Carlson (Über Geschwindigkeit und Grösse der Hefevermehrung in Würze, Biochemische Zeitschrift, Bd. 57, pp. 313–334, 1913) Table 11 (a) Using a graphing utility, draw a scatter plot of the data with time as the independent variable. (b) Using a graphing utility, build a logistic model from the data. (c) Using a graphing utility, graph the function found in part (b) on the scatter plot. (d) What is the predicted carrying capacity of the culture? (e) Use the function found in part (b) to predict the population of the culture at = t 19 hours. (continued)
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