Problem 59 - 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 \(=a \cdot b \cdot \pi\). \(\frac{(x+1)^{2}}{4}+\frac{(y-2)^{2}}{5}=1\)

Step-by-Step Solution

Answer

The area of the ellipse is \\( 2\sqrt{5}\pi \\\).

Step 1: Identify the Values of a and b

The standard form of an ellipse centered at \( (h, k) \) is given by \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). In this equation, \( rac{(x+1)^2}{4} + \frac{(y-2)^2}{5} = 1 \), we can identify \( a^2 = 4 \) and \( b^2 = 5 \). Therefore, \( a = 2 \) and \( b = \sqrt{5} \).

Step 2: Set Up the Area Formula

The area of the ellipse is given by the formula \( \text{Area} = a \cdot b \cdot \pi \). We know \( a = 2 \) and \( b = \sqrt{5} \). We can now substitute these values into the area formula.

Step 3: Calculate the Area

Substitute the values of \( a \) and \( b \) into the area formula: \[ \text{Area} = 2 \cdot \sqrt{5} \cdot \pi \] Simplifying, we get the area: \[ \text{Area} = 2\sqrt{5}\pi \]

Key Concepts

Ellipse Formula

An ellipse is a kind of stretched circle, but there is more to it than just being oval. At its core, the formula for finding the area of an ellipse is a multiplication of its semi-major and semi-minor axes with the mathematical constant pi (\(\pi\)). This gives us the formula:

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

Where \( a \) and \( b \) represent the lengths of the semi-major and semi-minor axes, respectively. These axes are essentially the 'radius' equivalents in an ellipse. In simpler terms, think about how wide and tall the ellipse is. The formula captures the concept that the area is wider when these axes get longer.It's crucial to understand how these values contribute to the overall area. Just like a circle, an ellipse's area is expressed in square units, showing us how much space is inside. Unlike a circle, which has one radius, the ellipse has two different measures, adding more complexity but also a fascinating twist to how we deal with areas.

Standard Form of an Ellipse

Recognizing the shape and form is vital, especially when dealing with mathematical equations involving ellipses. The standard form of an ellipse is:

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

In this form,

• \((h, k)\) represents the center of the ellipse.
• \(a^2\) and \(b^2\) are the denominators representing the square of the semi-major and semi-minor axes.

For example, with the equation \(\frac{(x+1)^{2}}{4}+\frac{(y-2)^{2}}{5}=1\), the ellipse's center is \((-1, 2)\). This shift from the origin shows how the equation adjusts to different ellipse placements.Identifying \(a\) and \(b\) is straightforward once the equation is in this form. The square roots of these denominators give us the original values of the axes. This transformation from square terms helps solve real-world problems, like finding how much area an elliptical shape covers.

Area Calculation

Now that we have the formula and the standard form in place, calculating an ellipse's area becomes a simple substitution task:

• Find \(a\) and \(b\) by taking the square root of the denominators.
• Substitute \(a\) and \(b\) into the formula \( \text{Area} = a \cdot b \cdot \pi \).

In our example, with \( a = 2 \) and \( b = \sqrt{5} \), the area calculation is:

• \( \text{Area} = 2 \cdot \sqrt{5} \cdot \pi \)

Subsequently simplifying yields \(2\sqrt{5}\pi\), which embodies the space occupied by the ellipse.Understanding this process helps us navigate through real-world applications where elliptical shapes appear, like planetary orbits or even racetracks. Mastering the substitution method alongside understanding how the formula works makes this task both straightforward and rewarding.