Problem 9 - Algebra 2

Question

Find an equation of an ellipse for each given height and width. Assume that the center of the ellipse is \((0,0) .\) $$ h=1 \mathrm{m}, w=3 \mathrm{m} $$

Step-by-Step Solution

Answer

The equation of the ellipse is \(0.44x^2 + 4y^2 = 1\).

Step 1: Identify the Given Values

From the problem, we know that the height (h) is 1m and the width (w) is 3m. The semi-major axis 'a' is half the width and the semi-minor axis 'b' is half the height in our case.

Step 2: Plug in the Values

As the center of the ellipse is at the origin (0,0), the equation becomes \((\frac{x^2}{(\frac{3}{2})^2}) + (\frac{y^2}{(\frac{1}{2})^2}) = 1\).

Step 3: Simplify

Simplify the equation to \((\frac{x^2}{2.25}) + (\frac{y^2}{0.25}) = 1\). Further simplification yields \(0.44x^2 + 4y^2 = 1\). This is the equation of the ellipse.

Key Concepts

Semi-Major Axis

The semi-major axis of an ellipse is an essential component in understanding its structure. It is the longest radius of the ellipse that extends from the center to the furthest edge. In simpler terms, it represents half of the longest diameter of the ellipse. For our exercise, the semi-major axis is one-half of the width given, meaning it is calculated as \( \frac{w}{2} \).

In our example, the width is 3 meters. Therefore, the semi-major axis, represented by \( a \), is calculated as:

• \( a = \frac{3}{2} = 1.5 \, \text{meters} \)

Understanding the semi-major axis is crucial for forming the standard equation of an ellipse.

Semi-Minor Axis

The semi-minor axis is another pivotal part of the ellipse's structure. It is the shortest radius extending from the center to the closest edge of the ellipse, effectively representing half the shortest diameter. For our specific case, the semi-minor axis is one-half of the height, which is given in the problem.

This height is 1 meter. So, the semi-minor axis \( b \) is calculated as:

• \( b = \frac{1}{2} = 0.5 \, \text{meters} \)

The semi-minor axis, together with the semi-major axis, helps define the size and shape of the ellipse in its mathematical equation.

Geometric Shapes

Ellipses are fascinating geometric shapes that belong to the family of conic sections. Unlike a perfect circle, which has equal radii, an ellipse has two distinct axes: the aforementioned semi-major and semi-minor axes.

This variation in axis length gives the ellipse its characteristic oval shape, making it a central figure in many mathematical applications such as physics and engineering. Geometric shapes like ellipses are defined by their symmetry and distinct mathematical properties. In the coordinate plane, an ellipse centered at the origin is often described by the formula \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Understanding these properties enhances learning about different geometric forms.

Quadratic Equations

Ellipses are described by a specific type of quadratic equation, which is a polynomial equation of degree two. Generally, a quadratic equation in its standard form is represented as \( ax^2 + bxy + cy^2 + dx + ey + f = 0 \). However, for ellipses situated at the origin, without rotation, the equation simplifies to:

\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

This form clearly showcases how the variables \( x \) and \( y \) relate to the semi-major (\( a \)) and semi-minor (\( b \)) axes. In our problem, after inserting the values for semi-major and semi-minor axes, we have:

• \( \frac{x^2}{1.5^2} + \frac{y^2}{0.5^2} = 1 \)

After simplifying this equation, you arrive at the standard quadratic equation for the ellipse. Grasping these quadratic relationships is essential for solving and understanding ellipses in algebra and beyond.