Problem 58 - 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 \(=a \cdot b \cdot \pi\). \(\frac{(x+6)^{2}}{16}+\frac{(y-6)^{2}}{36}=1\)

Step-by-Step Solution

Answer

The area of the ellipse is \(24\pi\).

Step 1: Identify the Semi-Axes

The equation of the ellipse is given as \( \frac{(x+6)^2}{16} + \frac{(y-6)^2}{36} = 1 \). This is in the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). From here, you can identify \( a^2 = 16 \) and \( b^2 = 36 \), which means \( a = 4 \) and \( b = 6 \).

Step 2: Apply the Area Formula

The formula for the area of an ellipse is given by \( \text{Area} = a \cdot b \cdot \pi \). Substitute the values \( a = 4 \) and \( b = 6 \) into the formula: \( \text{Area} = 4 \cdot 6 \cdot \pi \).

Step 3: Calculate the Area

Multiply the values as per the formula: \( 4 \cdot 6 = 24 \). Then, multiply by \( \pi \) to find the area: \( \text{Area} = 24\pi \). Therefore, the area of the ellipse is \( 24\pi \).

Key Concepts

Ellipse Equation

An ellipse is a smooth, symmetrical shape much like a stretched-out circle. You can describe it with an equation, a mathematical expression showing its characteristics. An ellipse's equation in standard form looks like this:

• \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)

In this equation, \((h, k)\) is the center of the ellipse. The terms under \((x - h)^2\) and \((y - k)^2\) indicate the distances squared from the center along the ellipse's axes. It's important to identify these terms correctly to understand the ellipse's size and shape. By comparing our example:

• \( \frac{(x+6)^2}{16} + \frac{(y-6)^2}{36} = 1 \)

you can see that the center is at \((-6, 6)\), and the values 16 and 36 help determine the axes' lengths. Understanding this equation is the first step to uncovering the area of the ellipse.

Semi-Major Axis

The semi-major axis is one of the critical dimensions of an ellipse, representing half of the longest length from one end of the ellipse to the other. In our ellipse equation:

• \( \frac{(x+6)^2}{16} + \frac{(y-6)^2}{36} = 1 \)

we identify \( b^2 = 36 \). By taking the square root, \( b = 6 \). Here, the value of \( b \) is associated with the semi-major axis. The longer axis stretches along the coordinate direction aligned with the term having the larger denominator. In this equation, the semi-major axis is aligned with the y-direction because 36 is greater than 16. The length of the semi-major axis is particularly significant because it reflects the ellipse's lengthiest span, fundamentally describing its overall shape.

Semi-Minor Axis

On the other hand, the semi-minor axis is half of the shortest length from one end of an ellipse to the other. In our standard form ellipse equation:

• \( \frac{(x+6)^2}{16} + \frac{(y-6)^2}{36} = 1 \)

we see \( a^2 = 16 \). Taking the square root gives \( a = 4 \), which denotes the semi-minor axis. This axis is shorter and is aligned with the x-direction here, again due to the smaller denominator, 16, compared to 36. The semi-minor axis is crucial in the calculation of the area, as we'll see in the next concept. It emphasizes the ellipse's narrowest width, providing a complete vision of its elongated form.

Area Formula

To find the area of an ellipse, we use a specific formula, which combines the lengths of both the semi-major and semi-minor axes:

• \( \text{Area} = a \cdot b \cdot \pi \)

Here, \( a \) is the semi-minor axis length, and \( b \) is the semi-major axis length. For our example, \( a = 4 \) and \( b = 6 \), so when these values are plugged into the formula, the area will be:

• \( \text{Area} = 4 \cdot 6 \cdot \pi = 24\pi \)

This result, \( 24\pi \), represents the total area contained within the ellipse's perimeter. The formula essentially multiplies all key dimensions together, incorporating \( \pi \) to account for the curved nature of the ellipse, similar to how it's used in the area of a circle. Thus, understanding this formula is key to solving problems related to the size of an ellipse.