Problem 14 - Algebra 2

Question

Find an equation of an ellipse for each given height and width. Assume that the center of the ellipse is \((0,0) .\) $$ h=8 \mathrm{ft}, w=2 \mathrm{ft} $$

Step-by-Step Solution

Answer

The equation of the ellipse with the given dimensions is \( x^2 + 4y^2 = 4 \)

Step 1: Determine the Width and Height

Given the height (h) is 8ft and width (w) is 2ft, we need to find half the height and width, as our formula uses these to represent 'a' and 'b'. So, 'a' is equal to half of width w, which is \( \frac{w}{2} = \frac{2}{2} = 1 \) and 'b' equal to half of height h, i.e \( \frac{h}{2} = \frac{8}{2} = 4 \)

Step 2: Insert Values into the Ellipse Formula

Using these half width 'a' and half height 'b' values, plug them into the general equation of an ellipse. We get the equation of the ellipse as follows: \[ \frac{x^2}{1} + \frac{y^2}{4} = 1 \] Or, it simplifies to \[ x^2 + 4y^2 = 4 \]

Key Concepts

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. It explores the concepts of points, lines, surfaces, and solids. An important part of geometry is understanding different figures and their properties. This specific exercise involves an ellipse, a shape commonly encountered in the study of geometry.

Ellipses are part of the conic sections, and understanding them involves recognizing their key features such as major and minor axes, foci, and center. A crucial part of learning geometry is comprehending how these elements come together to form various geometrical figures, like the ellipse in our exercise.

• The center, usually represented in coordinate geometry as \((h, k)\), is the central point of the ellipse. Here, it's \((0,0)\).
• The major axis is the longest diameter, and the minor axis is the shortest diameter of the ellipse.
• Learning geometry allows us to calculate distances, areas, and understand more about the spatial arrangement of figures.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. They include ellipses, parabolas, and hyperbolas. The study of conic sections in algebra and geometry exposes students to these crucial shapes and their properties.

In this exercise, we focus on an ellipse, one of these conic sections. An ellipse is defined by its two distinct radii:

• The semi-major axis, denoted as \(b\), which is half the greatest diameter of the ellipse.
• The semi-minor axis, denoted as \(a\), which is half the shortest diameter.

For the case at hand, the semi-minor axis is 1 foot, and the semi-major axis is 4 feet. Understanding conic sections provides insight into how these mathematical concepts can model real-world phenomena, such as planetary orbits that follow elliptical paths.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry to describe curves using equations. It gives us the ability to depict geometric figures through algebraic expressions and solve geometric problems.

This discipline is exceptionally useful for solving problems like finding the equation of an ellipse. In our example exercise, we are given an elliptical shape centered at the origin, \((0,0)\). Using the standard form of an ellipse equation, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) and \(b\) represent the ellipse's semi-axes, we can form its equation.

• When the center is at \((0,0)\), the equation reduces making calculations more direct.
• For our given measurements, we substitute \(a = 1\) and \(b = 4\) into the equation, resulting in the formula \(x^2 + 4y^2 = 4\).

Coordinate geometry is crucial for visualizing and verifying results and is foundational for advanced studies in mathematics and engineering.