Problem 92 - A Graphical Approach to College Algebra

Question

Solve each problem. The orbit of Venus is an ellipse, with the sun at one focus. An approximate equation for the orbit is $$\frac{x^{2}}{5013}+\frac{y^{2}}{4970}=1$$ where \(x\) and \(y\) are measured in millions of miles. (a) Approximate the length of the major axis. (b) Approximate the length of the minor axis.

Step-by-Step Solution

Answer

Major axis: ~141.6 million miles; Minor axis: ~141.0 million miles.

Step 1: Identify the Form of the Ellipse Equation

The given equation \( \frac{x^2}{5013} + \frac{y^2}{4970} = 1 \) is in the standard form of an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). In this form, \( a^2 \) and \( b^2 \) are the denominators, and if \( a^2 > b^2 \), then the ellipse is stretched more along the \( x \)-axis.

Step 2: Determine the Major and Minor Axes

Compare the denominators: \( 5013 \) and \( 4970 \). Since \( 5013 > 4970 \), the major axis is along the \( x \)-axis, making \( a^2 = 5013 \text{ million miles}^2 \) and \( b^2 = 4970 \text{ million miles}^2 \).

Step 3: Calculate the Length of the Major Axis

The length of the major axis is given by \( 2a \). Solve for \( a \): \( a = \sqrt{5013} \approx 70.8 \text{ million miles} \). Thus, the length of the major axis is approximately \( 2(70.8) = 141.6 \text{ million miles} \).

Step 4: Calculate the Length of the Minor Axis

The length of the minor axis is given by \( 2b \). Solve for \( b \): \( b = \sqrt{4970} \approx 70.5 \text{ million miles} \). Thus, the length of the minor axis is approximately \( 2(70.5) = 141.0 \text{ million miles} \).

Key Concepts

Major Axis

In the context of ellipses, the major axis is the longest diameter of the ellipse. It passes through the center and both foci of the ellipse. The major axis determines the overall stretch of the ellipse along its longest dimension. When examining the orbital path of celestial bodies like planets and comets, understanding the major axis helps astronomers predict the orbital track.
The major axis is an indicator of how elongated the ellipse is. In an equation like \( \frac{x^2}{5013} + \frac{y^2}{4970} = 1 \), the larger number under the variable, in this case, 5013, typically represents the semi-major axis squared. Here, because 5013 is greater than 4970, it tells us that the major axis is aligned along the x-axis.

• This axis extends from \(-a\) to \(a\), where \(a = \sqrt{5013}\).
• In this example, the length of the major axis is approximately 141.6 million miles.

Minor Axis

The minor axis is the shorter diameter of the ellipse. It is perpendicular to the major axis and stretches across the smaller dimension of the ellipse. While the major axis outlines the longest reach of the ellipse, the minor axis gives information about its shorter width.
In ellipse equations, the minor axis's length is derived from the smaller of the two denominators. In our equation \(\frac{x^2}{5013} + \frac{y^2}{4970} = 1 \), the minor axis is associated with the y-variable as 4970 is less than 5013.

• The length of the minor axis is given by \(2b\), where \(b = \sqrt{4970}\).
• The length is approximately 141.0 million miles.

Orbital Mechanics

Orbital mechanics is the study of the motions of celestial objects and is heavily reliant on understanding elliptical shapes, as many orbits take this shape. When a planet orbits a star, like Venus orbiting the sun, the orbit is not a perfect circle, but an ellipse.
An ellipse is defined by its two foci, and in the context of planets, one of these is typically the central sun or star. Understanding the major and minor axes of an orbit allows scientists to calculate orbital periods, velocities, and distances. These calculations are crucial not only for predicting planetary motion but also for spacecraft navigation.
For Venus’s orbit, the alignment and lengths of the major and minor axes determine how close or far it gets from the sun throughout its orbit.

Standard Form of Ellipse

The standard form of the ellipse is a mathematical arrangement that allows easy identification of the ellipse's properties, such as the length and orientation of its axes. It is written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) for ellipses centered at the origin.

• \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.
• If \(a^2 > b^2\), then the ellipse is stretched along the x-axis.
• If \(b^2 > a^2\), then the stretch is along the y-axis.

Our Venus orbit example, \(\frac{x^2}{5013} + \frac{y^2}{4970} = 1\), exemplifies how this form simplifies calculating the dimensions of the orbit. It allows direct comparison of the axes and determining which is the major and which is the minor.