If current rates of deforestation and fossil fuel...

If current rates of deforestation and fossil fuel consumption continue, then the amount of atmospheric carbon dioxide in parts per million (ppm) will be given by $f(x)=375 e^{00609 x},$ where $x=0$ corresponds to 2000 (a) What is the amount of carbon dioxide in $2003 ?$ In $2022 ?$ (b) In what year will the amount of carbon dioxide reach $500 \mathrm{ppm} ?$

If current rates of deforestation and fossil fuel consumption continue, then the amount of atmospheric carbon dioxide in parts per million (ppm) will be given by $f(x)=375 e^{00609 x},$ where $x=0$ corresponds to 2000
(a) What is the amount of carbon dioxide in $2003 ?$ In $2022 ?$
(b) In what year will the amount of carbon dioxide reach $500 \mathrm{ppm} ?$

Key Concepts

Exponential Growth Model

This model uses an exponential function to represent continuous growth processes where the rate of change is proportional to the current value. In the context of atmospheric carbon dioxide, it reflects how concentrations increase over time due to factors like deforestation and fossil fuel consumption.

Time Variable as an Offset

Here, the variable x represents the number of years after a baseline year (2000), simplifying the model by allowing x=0 to correspond to 2000. This makes it easier to translate specific years into the model's input and interpret the output as concentrations in those years.

Function Evaluation

Evaluating the exponential function f(x)=375e^(0.00609x) at specific values of x yields the predicted carbon dioxide concentrations for those years. For example, substituting x=3 for 2003 or x=22 for 2022 provides the ppm levels expected in those respective years.

Solving Exponential Equations with Logarithms

To find the time when the atmospheric carbon dioxide reaches a certain value (e.g., 500 ppm), one sets the exponential function equal to that value and solves for x. This typically involves isolating the exponential term and taking the natural logarithm of both sides to solve for the exponent, which represents the number of years after the base year.

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