Math Problem Statement

he amount of coal used by a small town over several decades is given in the table below.

YearTons of Coal

1930

87.4

1940

144.9

1950

153

1960

140.5

1970

131.6

1980

166.1

1990

254.2

a)   Perform regression to find a cubic polynomial that fits the data. Use years since 1930 as your inputs.

Solution

The cubic polynomial that fits the data is:

P(x)=0.00464x30.3827x2+9.0528x+87.6524P(x) = 0.00464x^3 - 0.3827x^2 + 9.0528x + 87.6524

where xx is the number of years since 1930.

Would you like further details on how the regression was performed or any clarification on the results?

Here are 5 related questions for further exploration:

  1. What does each coefficient in the cubic polynomial represent in this context?
  2. How well does this cubic model fit the data?
  3. Can we use this model to predict coal consumption beyond 1990?
  4. What other types of regression models could be applied to this dataset?
  5. How would the polynomial change if we added more recent data points?

Tip: When fitting a polynomial to data, always check the residuals to evaluate how well the model fits.

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Math Problem Analysis

Mathematical Concepts

Regression Analysis
Cubic Polynomial
Data Fitting

Formulas

Cubic Polynomial: P(x) = ax^3 + bx^2 + cx + d
Regression Formula for Polynomial Fitting

Theorems

Least Squares Method

Suitable Grade Level

Grades 11-12 or College Level