Problem 13 - Algebra 2

Question

Find an equation of an ellipse for each given height and width. Assume that the center of the ellipse is \((0,0) .\) $$ h=14 \text { yd, } w=28 $$

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{196} + \frac{y^2}{49} = 1\).

Step 1: Calculate half the height and width

First, since the coefficients 'a' and 'b' represent half the width and height of the ellipse, find these values by halving the given width and height. This gives \(a = \frac{28}{2} = 14\) and \(b = \frac{14}{2} = 7\).

Step 2: Substitute into the equation

Next, substitute these values for 'a' and 'b' into the general equation of an ellipse, getting \(\frac{x^2}{14^2} + \frac{y^2}{7^2} = 1\)

Step 3: Simplify the equation

Simplify the denominator of each fraction, yielding the final equation \(\frac{x^2}{196} + \frac{y^2}{49} = 1\)

Key Concepts

Center of Ellipse

The center of an ellipse is a fundamental part of its geometry. Imagine it as the very heart or the midpoint of the ellipse, from which all its dimensions are derived. For this exercise, the center is conveniently placed at the origin of the Cartesian coordinate system, at point the center, located at \((0,0)\).This simplifies calculations and helps to clearly define the symmetry of the ellipse. You can visualize the center as the pivot around which the entire shape of the ellipse is balanced. When the center of an ellipse is not at the origin, the calculations can become a bit more complex, as you will need to shift all points accordingly. But with the center at \((0,0)\), math turns wonderfully straightforward.In any problem asking for the equation of an ellipse, ensure you know the location of its center. It's the starting point for determining the orientation and placement of the ellipse in the plane.

Ellipse Dimensions

The dimensions of an ellipse are defined by its major and minor axes. These are the longest and shortest diameters that stretch across the ellipse. In our scenario:- The width (major axis length) is given as 28 yards.- The height (minor axis length) is 14 yards.However, for solving and using the general equation, we typically need the semi-major and semi-minor axes—essentially half of these lengths. By halving:- Semi-major axis (\(a\)): 14 yards- Semi-minor axis (\(b\)): 7 yardsThese halves represent how far the ellipse stretches out from the center in both directions. They are crucial when formulating the equation of the ellipse, outlining the boundaries for \(x\) and \(y\) values.

General Equation of an Ellipse

The general equation of an ellipse provides a standardized way to express its form in algebraic terms. This equation is given by:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. In the context of our exercise, we have:- \(a = 14\)- \(b = 7\)Substituting these values into the equation:\[\frac{x^2}{14^2} + \frac{y^2}{7^2} = 1\]This translates to:\[\frac{x^2}{196} + \frac{y^2}{49} = 1\]This final form expresses the ellipse's shape, size, and orientation. With these variables, you can describe any point \( (x, y) \) on the ellipse. Understanding this equation is essential, as it provides a clear mathematical representation of the physical ellipse visible in the graph, making it a key tool in both physics and engineering applications where ellipses frequently appear.