Problem 85 - 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. $$ 4 x^{2}+9 y^{2} \leq 36 $$

Step-by-Step Solution

Answer

The shaded region is an ellipse centered at (0,0) with area \(6\pi\) square feet.

Step 1: Rewrite the Inequality

The given inequality is \(4x^2 + 9y^2 \leq 36\). To understand the shape and location of the region, we want to rewrite this inequality in a form that is more familiar. Divide all terms by 36 to normalize it:\[\frac{x^2}{9} + \frac{y^2}{4} \leq 1.\]This represents an ellipse centered at the origin with semi-major axis length 3 along the x-axis and semi-minor axis length 2 along the y-axis.

Step 2: Graph the Inequality

An inequality involving \(\leq\) will include the area inside and on the boundary of the ellipse. To graph this, draw an ellipse centered at the origin. The ellipse extends 3 units in the positive and negative directions along the x-axis (since \(\sqrt{9} = 3\)) and 2 units in the positive and negative directions along the y-axis (since \(\sqrt{4} = 2\)). All points inside and on this ellipse satisfy the inequality.

Step 3: Calculate the Area of the Ellipse

The formula for the area of an ellipse is \(A = \pi \times a \times b\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. Substituting the values from our ellipse, we have:\[A = \pi \times 3 \times 2 = 6\pi.\]This gives the area of the region in square feet.

Key Concepts

inequality

An inequality is a mathematical expression that shows the relationship between two values, where one value is either greater than, less than, equal to, or not equal to the other. In algebra, inequalities are often used to represent a range of values that a particular variable can take.
For example, in the inequality \( 4x^2 + 9y^2 \leq 36 \), the symbol \( \leq \) means "less than or equal to." This signifies that the expression on the left must be either less than or equal to the number on the right, which in this case is 36.

• The inequality could represent a region of allowable solutions or outcomes in a graph, where \( x \) and \( y \) are coordinates.
• Solving inequalities often involves rearranging them, similar to solving equations, but considering the direction of the inequality sign.

Understanding inequalities is crucial as they provide a way to express constraints and limitations in mathematical models.

area of ellipse

The area of an ellipse is a fundamental concept in geometry. An ellipse resembles a stretched or flattened circle and is defined by two axes: the major axis and the minor axis. The major axis is the longest diameter of the ellipse, while the minor axis is the shortest diameter.
To find the area of an ellipse, we use the formula \( A = \pi \times a \times b \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis. The semi-major axis is half of the major axis, and the semi-minor axis is half of the minor axis.

• This formula shows that the area of the ellipse depends on both the lengths of these axes and the mathematical constant \( \pi \), which is approximately 3.14159.
• The calculation of the area of an ellipse allows us to measure the space within the boundary of the ellipse.

In the given problem, the semi-major axis is 3 and the semi-minor axis is 2, leading to an area of \( 6\pi\) square feet.

graphing inequalities

Graphing inequalities involves representing a range of solutions on a coordinate plane. For the inequality \( 4x^2 + 9y^2 \leq 36 \), graphing the equivalent equation \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) helps to visualize the boundary of the region described by the inequality. An ellipse is formed, and the area inside and on this ellipse boundary represents all solutions to the inequality.
Graphing inequalities can be simplified by taking a few steps:

• Normalize the inequality by dividing through by the constant term, if needed, to get it in its standard form.
• Identify the shape formed by the equation, which is an ellipse in this case.
• Sketch the shape on the coordinate plane, noting the lengths of the semi-major and semi-minor axes.

This visualization aids in understanding how changes in the variables \( x \) and \( y \) affect solutions to the inequality, showing which pairs of \( x, y \) satisfy the condition \( \leq \) in the problem.