Questions: The table lists future concentrations of a certain greenhouse gas in parts per billion ( ppb ), if current trends continue. (a) Find values for C and a so that f(x)=Cax models the data, where x is years after 2000. (b) Estimate the gas concentration in 2019. Year 2000 2005 2010 2015 2020 Gas (ppb) 0.86 1.26 1.86 2.73 4.01 (a) It is convenient to first find the value of C in f(x)=Cx. Determine the values of C and a . C=0.86 a ≈ 1.08 (Round to the nearest hundredth.) (b) The estimated gas concentration in 2019 will be approximately ppb. (Round to the nearest hundredth.)

Solution Steps

To solve this problem, we need to model the given data using an exponential function of the form \( f(x) = Ca^x \), where \( x \) is the number of years after 2000. First, we determine the initial value \( C \) as the gas concentration in the year 2000. Then, we calculate the growth factor \( a \) by using the data points provided. Once we have \( C \) and \( a \), we can estimate the gas concentration for the year 2019 by substituting \( x = 19 \) into the model.

The initial concentration of the greenhouse gas in the year 2000 is given as \( 0.86 \) ppb. Therefore, we have: \[ C = 0.86 \]

To find the growth factor \( a \), we use the formula: \[ a = \left( \frac{f(x_2)}{f(x_1)} \right)^{\frac{1}{x_2 - x_1}} \] Using the concentrations from the years 2000 and 2020: \[ a = \left( \frac{4.01}{0.86} \right)^{\frac{1}{20}} \approx 1.0800 \]

To estimate the gas concentration in 2019, we substitute \( x = 19 \) into the model \( f(x) = Ca^x \): \[ f(19) = 0.86 \cdot (1.08)^{19} \approx 3.7129 \] Rounding this value to two decimal places gives: \[ f(19) \approx 3.71 \]

Final Answer