368 CHAPTER 5 Exponential and Logarithmic Functions Fitting a Logarithmic Function to Data Jodi, a meteorologist, is interested in finding a function that explains the relation between the height of a weather balloon (in kilometers) and the atmospheric pressure (measured in millimeters of mercury) on the balloon. She collects the data shown in Table 10. (a) Using a graphing utility, draw a scatter plot of the data with atmospheric pressure as the independent variable. (b) It is known that the relation between atmospheric pressure and height follows a logarithmic model. Using a graphing utility, build a logarithmic model from the data. (c) Graph the logarithmic function found in part (b) on the scatter plot. (d) Use the function found in part (b) to predict the height of the weather balloon if the atmospheric pressure is 560 millimeters of mercury. Solution EXAMPLE 2 Atmospheric Pressure, p Height, h 760 0 740 0.184 725 0.328 700 0.565 650 1.079 630 1.291 600 1.634 580 1.862 550 2.235 Table 10 Figure 65 TI-84 Plus CE 525 20.2 775 2.4 Figure 63 0 21 40,000 7 Figure 64 Exponential model using Desmos* (d) See Figure 63 for the graph of the exponential function of best fit on a TI-84 Plus CE. Figure 64 shows the exponential model using Desmos. (e) Let = t 10 in the function found in part (c). The predicted value of the account after 10 years is = = ≈ ⋅ A A e e 19,820.43 $45,047 kt 0 0.08210 10 (f) The value of = = k 0.08210 8.210% represents the annual growth rate of the account. It represents the rate of interest earned, assuming the account is growing continuously. Now Work PROBLEM 1 2 Build a Logarithmic Model from Data Some relations between variables follow a logarithmic model. (a) Enter the data into the graphing utility, and draw the scatter plot. See Figure 65. (b) A graphing utility fits the data in Table 10 to a logarithmic function of the form = + y a b x ln by using the LOGarithm REGression option. Figure 66 on the next page shows the result on a TI-84 Plus CE. The logarithmic model from the data is ( ) = − h p p 45.7863 6.9025 ln where h is the height of the weather balloon and p is the atmospheric pressure. Notice that r is close to 1, indicating a good fit. *For this result in Desmos to agree precisely with the result of a TI-84 Plus CE, the “Log Mode” option must be selected. Consult the help feature in Desmos for more information about this option.
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