SECTION 5.9 Building Exponential, Logarithmic, and Logistic Models from Data 371 4. Chemistry A chemist has a 100-gram sample of a radioactive material. He records the amount of radioactive material every week for 7 weeks and obtains the following data: Week Weight (in grams) 0 1 2 3 4 5 6 100.0 88.3 75.9 69.4 59.1 51.8 45.5 (a) Using a graphing utility, draw a scatter plot with week 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 ( ) = A t A e . kt 0 (d) Graph the exponential function found in part (b) or (c) on the scatter plot. (e) From the result found in part (b), determine the half-life of the radioactive material. (f) How much radioactive material will be left after 50 weeks? (g) When will there be 20 grams of radioactive material? 5. Milk Production The data in the table below represent the number of dairy farms (in thousands) and the amount of milk produced (in billions of pounds) in the United States for various years. Year Dairy Farms (thousands) Milk Produced (billion pounds) 1980 334 128 1985 269 143 1990 193 148 1995 140 155 2000 105 167 2005 78 177 2010 63 193 2015 44 209 2020 64 223 Source: National Agricultural Statistics Services. (a) Using a graphing utility, draw a scatter plot of the data with the number of dairy farms as the independent variable. (b) 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) In 2008, there were 67 thousand dairy farms in the United States. Use the function in part (b) to predict the amount of milk produced in 2008. (e) The actual amount of milk produced in 2008 was 190 billion pounds. How does your prediction in part (d) compare to this? 2. Tesla, Inc. Revenue The data in the table below represent annual revenue of Tesla, Inc. from 2010 to 2018. Year Revenue ($ Billion) 2010 ( ) = x 0 0.12 2011 ( ) = x 1 0.20 2012 ( ) = x 2 0.41 2013 ( ) = x 3 2.01 2014 ( ) = x 4 3.20 2015 ( ) = x 5 4.05 2016 ( ) = x 6 7.00 2017 ( ) = x 7 11.76 2018 ( ) = x 8 21.46 2019 ( ) = x 9 24.58 2020 ( ) = x 10 31.54 2021 ( ) = x 11 53.82 2022 ( ) = x 12 81.46 2023 ( ) = x 13 96.77 Source: Tesla, Inc. (a) Using a graphing utility, draw a scatter plot of the data using 0 for 2010, 1 for 2011, and so on, 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 ( ) = A t A 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 Tesla’s revenue in 2025. (f) Interpret the meaning of k in the function found in part (c). 3. Advanced-stage Breast Cancer The data in the table below represent the percentage of patients who have survived after diagnosis of advanced-stage breast cancer at 6-month intervals of time. Time after Diagnosis (years) Percentage Surviving 0.5 95.7 1 83.6 1.5 74.0 2 58.6 2.5 47.4 3 41.9 3.5 33.6 Source: Cancer Treatment Centers of America (a) Using a graphing utility, draw a scatter plot of the data with time after diagnosis 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 ( ) = A t A e . kt 0 (d) Graph the exponential function found in part (b) or (c) on the scatter plot. (e) Use the model to predict the percentage of patients diagnosed with advanced-stage cancer who survive for 4 years after initial diagnosis? (f) Interpret the meaning of k in the function found in part (c).
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