Problem 29 - Contemporary Precalculus

Question

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$3 x^{2}+4 y^{2}=12$$

Step-by-Step Solution

Answer

Answer: The area of the ellipse is \(2\pi\sqrt{3}\).

Step 1: Rewrite the equation

The given ellipse has the equation \(3x^2+4y^2=12.\) To rewrite it in the form \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\) we need to divide both sides by the constant 12: $$\frac{3x^2}{12}+\frac{4y^2}{12}=1.$$ Simplify: $$\frac{x^2}{4}+\frac{y^2}{3}=1.$$

Step 2: Identify `a` and `b`

Now we can compare the equation \(\frac{x^2}{4}+\frac{y^2}{3}=1\) to the standard form \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.\) We see that \(a^2=4\) and \(b^2=3.\) Therefore, we have \(a=2\) and \(b=\sqrt{3}.\)

Step 3: Find the area of the ellipse

Using the formula for the area of an ellipse, \(area=\pi ab,\) we plug in the values of \(a=2\) and \(b=\sqrt{3}\) to get: $$area=\pi(2)(\sqrt{3})$$ So the area of the given ellipse is: $$area=2\pi\sqrt{3}.$$

Key Concepts

Precalculus

Precalculus is an essential foundation for understanding more advanced mathematics, like calculus. It involves studying algebra, geometry, and trigonometry. One key part of precalculus is learning to manipulate equations, which will help solve more complex problems. Knowing how to work with different mathematical expressions prepares you for calculus and other higher-level math topics.

With precalculus, students focus on various functions and their properties. Evaluating and graphing functions, like ellipses, becomes easier once you understand transformations and how they adjust equations. In our case, rewriting an ellipse's equation in its standard form is an essential step to finding the necessary elements, such as semi-major and semi-minor axes. These skills are fundamental before transitioning into calculus, where derivatives and integrals are introduced.

A strong grasp of precalculus empowers students to approach calculus with more confidence and comprehension. It serves as the bridge between basic arithmetic and the world of advanced mathematics, providing the necessary tools to tackle real-world mathematical problems.

Standard Form of Ellipse Equation

Ellipses are intriguing shapes that appear frequently in geometry and precalculus. An ellipse is defined as a set of points where the sum of the distances from two fixed points, known as foci, is constant.

The standard form of an ellipse's equation is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. This form allows for easy identification and manipulation of the ellipse's properties.

To convert a general ellipse equation, like \(3x^2 + 4y^2 = 12\), into this standard form, dividing through by the constant achieves the desired expression. Simplifying gives \(\frac{x^2}{4} + \frac{y^2}{3} = 1\), as seen in our example.

This helps us identify \(a^2 = 4\) and \(b^2 = 3\), leading to \(a = 2\) and \(b = \sqrt{3}\). This step is crucial for solving further questions about the ellipse, such as its area. Understanding how to find this standard form is key when working with ellipses in various mathematical and real-world applications.

Area of Ellipse Formula

Finding the area of an ellipse can seem daunting, but once you grasp the formula, it's straightforward. Unlike circles, whose area is found using \(\pi r^2\), ellipses require both semi-major and semi-minor axes for calculation.

The formula to find the area of an ellipse is \[ \text{Area} = \pi ab \] where \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively. This formula stems from calculus, but precalculus provides the tools needed to apply it without delving into derivatives or integrals.

For the ellipse \(3x^2 + 4y^2 = 12\), after converting it to standard form, we determined \(a = 2\) and \(b = \sqrt{3}\). Plugging these values into the formula gives us \(\pi \times 2 \times \sqrt{3} = 2\pi\sqrt{3}\).

This method allows quick and accurate computation of an ellipse's area, useful in various fields such as engineering, physics, and astronomy. Understanding this formula and knowing how to identify \(a\) and \(b\) is crucial when working with ellipses.