Energy The number of coal mines C in the United States from...

Energy The number of coal mines $C$ in the United States from 1980 to 2004 can be approximated by the equation $C=0.232 t^{3}-2.11 t^{2}-261.8 t+5699,$ for $0 \leq t \leq 24$ , where $t$ is the year, with $t=0$ corresponding to 1980 . (Source: U.S. Energy Information Administration) (a) Use a graphing utility to graph the model over the domain. (b) Find the number of mines in $1980 .$ Use the Remainder Theorem to find the number of mines in 1990 . (c) Could you use this model to predict the number of coal mines in the United States in the future? Explain.

Energy The number of coal mines $C$ in the United States from 1980 to 2004 can be approximated by the equation $C=0.232 t^{3}-2.11 t^{2}-261.8 t+5699,$ for $0 \leq t \leq 24$ , where $t$ is the year, with $t=0$ corresponding to 1980 . (Source: U.S. Energy Information Administration)
(a) Use a graphing utility to graph the model over the domain.
(b) Find the number of mines in $1980 .$ Use the Remainder Theorem to find the number of mines in 1990 .
(c) Could you use this model to predict the number of coal mines in the United States in the future? Explain.

Key Concepts

Polynomial Modeling

This concept involves representing real-world data and scenarios using polynomial functions. Such models capture the relationships between variables, allowing for analyses of trends over time and predictions within the scope of the data. It is fundamental in fields like economics, physics, and biology for modeling complex systems.

Graphing Functions

Graphing is the process of visually representing functions on a coordinate plane to observe their behavior, including trends, turning points, and intercepts. It helps in understanding the overall shape of the model and provides visual confirmation of properties such as continuity and domain restrictions.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is equal to f(a). This simplifies finding the value of a polynomial at a specific point and is an essential tool in polynomial division and evaluation without the need for complete factorization.

Extrapolation and Prediction Limitations

This concept distinguishes between predicting values within the range of known data (interpolation) versus those beyond it (extrapolation). Extrapolation may lead to unreliable results since it assumes that established trends continue indefinitely. This caution is important when using models to predict future outcomes where system behavior may change.

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