Question

The area of an ellipse whose semi-minor and semi-major axes are \(r\) and \(r+1\) respectively is given below. Which model best fits the data? $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Radius, } r & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline \text { Area } A & 2 \pi & 6 \pi & 12 \pi & 20 \pi & 30 \pi & 42 \pi \\\\\hline\end{array}$$ (A) \(A=\pi r+2(B) A=\pi r(r+1)\) (C) \(A=\pi(1+r)^{t}\) (D) \(A=r^{2}+\pi r+2\)

Short answer

The best fit is \(A=\pi r(r+1)\) if this model can generate the areas given in the problem for all \( r \)

Step-by-step solution

Examine the given models

Evaluate the given models by substituting different values of \( r \) into each model to see if it can generate the respective areas.

Test model A

Substitute \( r = 1 \) into Model A: \( A = \pi \times 1 + 2 = \pi + 2 \). This does not equal to \( 2 \pi \) given in the problem, so Model A is incorrect.

Test model B

Substitute \( r = 1 \) into Model B: \( A = \pi \times 1 \times (1+1) = 2 \pi \). Now try another value \( r = 2 \), then \( A = \pi \times 2 \times (2+1) = 6 \pi \). Perform similar checks for all other given values of \( r \). If each evaluated value matches with the corresponding area in the problem, Model B is correct.

Check remaining models

If Model B doesn't fit all data points, continue with Models C and D. Use the same process as in Steps 2 and 3.

Conclusion

The model which correctly calculates the area of ellipse for all given \( r \) values is the best fit for the data.

Key concepts

Algebraic Modeling

Algebraic modeling is a powerful mathematical approach that helps us translate real-world phenomena into algebraic equations. It allows us to make predictions, analyze trends, and solve problems by defining variables and constructing equations to represent relationships between them.

For instance, when dealing with geometric shapes such as an ellipse, algebraic modeling involves understanding the attributes of the shape and then expressing its area, perimeter, or other properties in a formulaic way. In our exercise, the area of an ellipse is being modeled as a function of its semi-minor and semi-major axes. Different algebraic models (A, B, C, and D) are proposed and need to be evaluated against the provided data to identify which one accurately represents the relationship between the radius and the area of the ellipse.

By systematically substituting values into the proposed models and checking against the known areas, the correct algebraic model becomes evident, showcasing the practical utility of algebraic modeling in solving geometric problems.

Geometric Shapes in Algebra

In algebra, geometric shapes are often represented through equations that encapsulate their fundamental properties. These algebraic representations are essential for performing calculations such as area, volume, and other spatial relationships.

The ellipse is a prime example of how geometric shapes are modeled in algebra. Defined by its semi-major and semi-minor axes, the area of an ellipse is formulaically given by \( A = \pi a b \), where \( a \) is the length of the semi-major axis and \( b \) is the length of the semi-minor axis. This formula is derived from the circle's area formula \( A = \pi r^2 \), with the ellipse being considered as a 'stretched' circle. In our exercise, we explore algebraic models aiming to establish a relationship that correctly expresses the area of ellipses with different radii, highlighting the importance of geometric understanding in algebraic expressions.

Pi in Algebra

The number \(\pi\) is a fundamental component in the algebraic expression of circular and elliptical shapes. In algebra, \(\pi\) is used to connect the linear dimensions of a shape with its area or volume. It serves as the bridge between algebra and geometry, translating linear measurements into spatial properties.

Within the exercise provided, \(\pi\) appears in every model, emphasizing its indispensable role in determining the area of an ellipse. The presence of \(\pi\) underscores the circular nature of an ellipse; in essence, it is a constant that relates the semi-minor and semi-major axes to the shape's overall space. In understanding how \(\pi\) functions algebraically, students can better appreciate why we multiply it by the axes measurements to calculate the area of an ellipse and why it is ingrained in each of the proposed algebraic models for the ellipse area calculation.