Problem 87 - 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. $$ \frac{(x-1)^{2}}{25}+\frac{(y+2)^{2}}{16} \leq 1 $$

Step-by-Step Solution

Answer

The area of the shaded region is \(20\pi\) square feet.

Step 1: Identify the Inequality Type

The given inequality \( \frac{(x-1)^{2}}{25}+\frac{(y+2)^{2}}{16} \leq 1 \) represents an ellipse centered at \((1, -2)\) with semi-major axis 5 (since \( \sqrt{25} = 5 \)) along the x-axis and semi-minor axis 4 (since \( \sqrt{16} = 4 \)) along the y-axis.

Step 2: Confirm the Center, Axes, and Region

Verify the center of the ellipse at \((1, -2)\), with major axis length 10 feet along the x-axis and minor axis length 8 feet along the y-axis. The inequality includes the boundary, hence the region includes all the points inside and on the ellipse.

Step 3: Calculate the Area of the Ellipse

The formula for the area of an ellipse is \( A = \pi \times a \times b \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis. Substitute the values \( a = 5 \) and \( b = 4 \) to find the area: \[ A = \pi \times 5 \times 4 = 20\pi. \]

Step 4: Interpret the Results

The solution requires shading the interior and the boundary of the ellipse on the xy-plane. The calculated area is the total region allowed by the inequality, covering 20\( \pi \) square feet.

Key Concepts

Inequalities

Inequalities in mathematics define a relationship where an expression is not strictly equal but rather greater or less in comparison to another. For the inequality \( \frac{(x-1)^{2}}{25}+\frac{(y+2)^{2}}{16} \leq 1 \), it means that any point \((x, y)\) in this relation is either on the ellipse's border or inside it. The inequality takes the form of an ellipse equation \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), but with '\( \leq \)' instead of '\( = \)', signifying inclusion of all interior points. This additional region involves all points within the closed shape (the ellipse) outlined by the equation's equality boundary, rather than just the points that satisfy the boundary equation. Therefore, the task involves determining the set of coordinates that lie within—and on—the ellipse, as the inequality's "\( \leq \)" indicates allowance for equality, ensuring the boundary is included.

Area of an Ellipse

Finding the area of an ellipse is a matter of using its unique area formula. Unlike circles with a simple area formula \( \pi r^2 \), ellipses require another approach due to their elongated shape. The area \( A \) of an ellipse is given by the formula: \( A = \pi \times a \times b \). Here, \( a \) represents the semi-major axis, and \( b \) represents the semi-minor axis of the ellipse. Both axes meet at the center of the ellipse and extend outwards towards its widest and narrowest points. In this scenario, the semi-major axis \( a = 5 \) and the semi-minor axis \( b = 4 \). Therefore, the area is calculated as \( 20\pi \), capturing the full extent of the ellipse's spread in square units. This mathematical model provides a straightforward measure of space taken by shapes that do not necessarily conform to regular round figures.

Coordinate Geometry

Coordinate geometry serves as a bridge between algebra and geometry that uses a coordinate plane to explore geometric shapes through equations. In this exercise, coordinate geometry takes form using the given equation of an ellipse. The equation \( \frac{(x-1)^{2}}{25}+\frac{(y+2)^{2}}{16} \leq 1 \) conveys a geometric figure centered on the coordinate plane at point \((1, -2)\). Coordinate axes allow representation of each point \((x, y)\) that satisfies the ellipse’s boundary or internal region.

• The point \((h, k)\), which is \((1, -2)\) here, helps to translate the center of the ellipse from the origin to a new position involving positive or negative shifts on the axes.
• The terms inside the fractions represent distances squared from the center point along the x and y directions, respectively.

By learning coordinate geometry, you understand how to visualize algebraic equations as objects with specific shapes and sizes on the plane, helping you take abstract concepts to tangible drawings even in mind's eye.