Because of the combustion of fossil fuels, the concentration...

Because of the combustion of fossil fuels, the concentration of carbon dioxide in the atmosphere is increasing. Research indicates that this will result in a greenhouse effect that will change the average global surface temperature. Assuming a vigorous expansion of coal use, the future amount $A(t)$ of atmospheric carbon dioxide concentration can be approximated (in parts per million) by $$ A(t)=-\frac{1}{2400} t^{3}+\frac{1}{20} t^{2}+\frac{7}{6} t+340 $$ where $t$ is in years, $t=0$ corresponds to $1980,$ and $0 \leq t \leq 60 .$ Use the graph of $A$ to estimate the year when the carbon dioxide concentration will be 400 .

Because of the combustion of fossil fuels, the concentration of carbon dioxide in the atmosphere is increasing. Research indicates that this will result in a greenhouse effect that will change the average global surface temperature. Assuming a vigorous expansion of coal use, the future amount $A(t)$ of atmospheric carbon dioxide concentration can be approximated (in parts per million) by
$$
A(t)=-\frac{1}{2400} t^{3}+\frac{1}{20} t^{2}+\frac{7}{6} t+340
$$
where $t$ is in years, $t=0$ corresponds to $1980,$ and $0 \leq t \leq 60 .$ Use the graph of $A$ to estimate the year when the carbon dioxide concentration will be 400 .

Key Concepts

Time Translation

Time translation refers to relating the mathematical variable (often a proxy for time) back to actual calendar dates or real-world periods. In models where time is shifted or scaled (for instance, having t = 0 represent a specific base year), it is crucial to correctly convert the computed time value back into the corresponding calendar year for accurate interpretation and communication of results.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent and simulate real-world phenomena. In this context, a polynomial function is used to approximate the behavior of atmospheric carbon dioxide concentration over time, allowing predictions about future conditions based on current and past trends.

Polynomial Functions

Polynomial functions are algebraic expressions made up of terms in the form of constants multiplied by powers of a variable. They are particularly useful in modeling scenarios where the underlying process shows trends such as acceleration or deceleration, as the coefficients and degree of the polynomial capture the rate and nature of these changes.

Graph Interpretation

Graph interpretation is the process of analyzing a function’s graphical representation to extract meaningful information. It allows one to visually locate where the function reaches specific values, assess trends, and estimate outcomes even when an explicit algebraic solution is challenging or unwieldy.

Root Finding

Root finding is the mathematical process of determining the values at which a function equals a given constant. In applied contexts, this helps identify critical points, such as the time when a modeled quantity reaches a particular threshold, by setting the function equal to that value and solving for the variable.

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