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An ellipse is a fascinating geometric shape, often looked at as an elongated circle. It has two axes, namely the major axis and the minor axis. Imagine stretching a circle into an oval shape; that's your ellipse. The two axes have distinct names and uses:

• The **major axis** is the longest diameter, stretching across the widest part of the ellipse.
• The **minor axis** runs perpendicular to the major axis, spanning the shortest part across the ellipse.

Understanding ellipses is crucial in various fields like astronomy and engineering. For example, the orbits of planets are elliptical. The formula for the area of an ellipse, given its semi-major axis, "a," and semi-minor axis, "b," is: \[A = \pi a b\]This formula shows how the ellipse's shape affects its area. Changes to either axis length can significantly alter the computed area. Using this concept, you'll grasp how ellipses transform with different axis measurements and sized adjustments.

Percent change is a way to express the amount of change over a particular period or due to a specific modification in terms of a percentage. It's a practical tool for comparing old and new values, commonly used in economics, statistics, and many real-world tasks.To calculate percent change, you'll use the formula:\[\%\, Change = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\]In our ellipse context, this formula is applied to find the percentage change in the ellipse's area after changes in its axes.**Steps to Calculate:**

• Determine the original and new values you are comparing.
• Subtract the original value from the new value.
• Divide this by the original value to find the change as a fraction of the original.
• Multiply by 100 to convert the fraction into a percentage.

Regardless of the context, understanding percent change can help you notice and understand shifts and trends, such as economic growth or, in our situation, area changes due to axis increases. It's a versatile and essential mathematical tool.

Calculating area is a key part of understanding geometry and shapes. The area tells us how much two-dimensional space a shape covers. Whether calculating for rectangles, circles, or any special shapes like ellipses, it's essential to have the correct formula.**For Ellipses:**

• The area formula: \(A = \pi a b\) is specific to ellipses and is very distinct from other shapes like circles \(A = \pi r^2\) or rectangles \(A = \text{length} \times \text{width}\).
• This formula uses two parameters: the semi-major axis \(a\) and the semi-minor axis \(b\).

To apply this effectively:

• Measure the lengths of axes correctly.
• Understand how changes in these lengths affect overall area.

When the axes change, even slightly, the area can change significantly, as demonstrated in the exercise. It's a great example of how small modifications in dimensions impact the computed area. Understanding this concept through examples helps illustrate how geometric properties are deeply connected to algebraic expressions and can be used to predict changes in real-world scenarios.