Problem 86 - College Algebra with Modeling and Visualization

Question

Shade the region in the xy-plane that satisfies the given inequality. Find the area of this region if units are in feet. $$ 9 x^{2}+y^{2} \leq 9 $$

Step-by-Step Solution

Answer

The area of the shaded region is \(3\pi\) square feet.

Step 1: Identify the Equation of the Ellipse

First, rewrite the inequality in standard form for an ellipse. Divide every term by 9 to obtain:\[ \frac{x^2}{1} + \frac{y^2}{9} \leq 1 \]This corresponds to an ellipse centered at the origin (0,0) with semi-major axis along the y-axis and semi-minor axis along the x-axis.

Step 2: Determine the Axes Length

The semi-minor axis has length 1 (since \(x^2/1\)) and the semi-major axis has length 3 (\(y^2/9\), so \(\sqrt{9} = 3\)). Hence, the ellipse stretches 1 unit in the x-direction and 3 units in the y-direction.

Step 3: Calculate the Area of the Ellipse

The area \(A\) of an ellipse is given by the formula:\[ A = \pi \times a \times b \]where \(a\) is the length of the semi-minor axis, and \(b\) is the length of the semi-major axis. Here, \(a = 1\) and \(b = 3\), so:\[ A = \pi \times 1 \times 3 = 3\pi \text{ square feet} \]

Step 4: Verify the Shaded Region

Since the given inequality involves a less than or equal sign (\(\leq\)), the region inside or on the ellipse is shaded. This confirms that the area calculated in Step 3 is for the shaded region in the inequality.

Key Concepts

Area of an Ellipse

An ellipse is a beautiful and unique shape, and calculating its area is a fundamental concept in geometry. Unlike a circle, which has only one radius, an ellipse has two important dimensions - the semi-major and semi-minor axes. The area of an ellipse can be calculated using the formula:

• \( A = \pi \times a \times b \)

where \(a\) is the length of the semi-minor axis and \(b\) is the length of the semi-major axis.
To better understand, think of the ellipse as a 'stretched' circle. The stretching is determined by the lengths of the axes. In this exercise, the semi-minor axis is 1 unit long and the semi-major axis is 3 units long.
If we plug these values into the formula, we get:

• \( A = \pi \times 1 \times 3 = 3\pi \text{ square feet} \)

This tells us that the area enclosed by the ellipse, in this case, is \(3\pi\) square feet. This calculated area is applicable if we consider the region defined by the inequality \(9x^2 + y^2 \leq 9\).

Semi-Major and Semi-Minor Axes

The two axes of an ellipse are essential for understanding its geometry. The semi-major axis is the longer one, and the semi-minor axis is the shorter one. These axes determine the ellipse's shape and orientation.
For the given inequality \(9x^2 + y^2 \leq 9\), we first rewrite it in the standard ellipse form: \(\frac{x^2}{1} + \frac{y^2}{9} \leq 1\).
Here,

• The semi-minor axis has a length of 1.
• The semi-major axis has a length of 3.

You can observe that the semi-major axis is aligned along the y-axis in this case. This is because the coefficient under \(y^2\) (9) is larger than that under \(x^2\) (1), resulting in the ellipse stretching more in the y-direction compared to the x-direction.
Understanding the axes gives you a clear picture of the ellipse's dimensions and helps in sketching it accurately.

Graphical Representation of Inequalities

Graphically representing inequalities can help visualize solutions to problems quite effectively. In the context of this exercise, we are dealing with an inequality involving an ellipse: \(9x^2 + y^2 \leq 9\).
This inequality represents all the points \((x, y)\) that are inside or on the boundary of the ellipse centered at the origin with semi-major axis 3 and semi-minor axis 1.
To graphically represent this inequality, you would first draw the ellipse using the axes lengths. The shading approach is crucial here, and since we have a "less than or equal to" (\(\leq\)) sign, you would shade the entire region inside the ellipse including the boundary.
Shading visually confirms the solution of the inequality. It helps identify any point inside or on the ellipse as a solution point, hence contributing to better understanding. Practicing this kind of graphical representation makes it easier to tackle more complex inequalities.