Problem 57 - College Algebra with Corequisite Support

Question

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area \(=a \cdot b \cdot \pi\). \(\frac{(x-3)^{2}}{9}+\frac{(y-3)^{2}}{16}=1\)

Step-by-Step Solution

Answer

The area of the ellipse is \(12\pi\).

Step 1: Identify the values of a and b

The standard form of the equation of an ellipse is \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. In the given equation \(\frac{(x-3)^{2}}{9}+\frac{(y-3)^{2}}{16}=1\), we identify \(a^{2}=9\) and \(b^{2}=16\). Thus, \(a=3\) and \(b=4\).

Step 2: Calculate the area of the ellipse

The area of an ellipse is calculated using the formula \(\text{Area} = a \cdot b \cdot \pi\). Substituting the identified values \(a = 3\) and \(b = 4\), the area is \(3 \cdot 4 \cdot \pi = 12\pi\).

Key Concepts

Ellipse Equation

An ellipse is a geometric shape that looks like an elongated circle. Its mathematical representation is key to many problems in geometry. The standard form of the equation for an ellipse is \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\). Here, \((h, k)\) represents the center of the ellipse. Meanwhile, \(a\) and \(b\) are crucial values known as the semi-major and semi-minor axes, respectively.
The equation shows how the ellipse extends along the x and y axes from its center point. Understanding this standard form is essential for working with ellipses, as it allows you to identify the geometry's dimensions and orientation with ease.

Semi-major Axis

The semi-major axis, denoted as \(a\), is one of the ellipse's prominent features. It is the longest radius that extends from the center to the edge of the ellipse along the major axis.
In the standard equation \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\), \(a^2\) is the denominator of the x-component when the semi-major axis is oriented horizontally. This positioning means the ellipse is wider than it is tall.

• When \(a > b\), the ellipse is stretched along the x-axis.
• If \(a < b\), the semi-major axis aligns with the y-axis, and the ellipse is taller than it is wide.

Choosing the correct semi-major axis is important for correctly describing or constructing an ellipse from its equation.

Semi-minor Axis

The semi-minor axis, represented as \(b\), contrasts with the semi-major axis; it represents the shortest radius extending from the ellipse's center to its edge. This axis is significant for defining the ellipse's dimensions.
In our ellipse equation \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\), \(b^2\) appears in the denominator of the y-component when the ellipse stretches horizontally beside the semi-major axis. This means:

• \(b\) is less than \(a\) when the ellipse is horizontally oriented.
• If \(b > a\), the semi-minor axis becomes vertical, making the ellipse taller.

This axis, while shorter, is crucial for understanding how compact or stretched an ellipse is in its shape.

Ellipse Area Formula

The area of an ellipse can be calculated using the formula: \(\text{Area} = a \cdot b \cdot \pi\). This formula is vital for numerous applications in mathematics and science, as it helps determine the size of this shape.
Here's how the formula works:

• \(a\) is the length of the semi-major axis, measuring the longest distance from the center.
• \(b\) is the length of the semi-minor axis, measuring the shortest distance from the center.

Multiplying these gives a measure of the rectangle that could encapsulate the ellipse, while multiplying by \(\pi\) accounts for the round shape.
Using the factor of \(\pi\) assures the formula accurately represents the elliptical area, similar to how \(\pi\) relates to the area of a circle. This formula emphasizes the importance of the semi-major and semi-minor axes in understanding the geometrical properties of an ellipse.