Problem 204 - Calculus 3

Question

The area of an ellipse with axes of length 2\(a\) and 2\(b\) is given by the formula \(A=\pi a b .\) Approximate the percent change in the area when \(a\) increases by 2\(\%\) and \(b\) increases by 1.5\(\%.\)

Step-by-Step Solution

Answer

The area increases by approximately 3.5\%.

Step 1: Understand the problem

The problem asks for the approximate percent change in the area of an ellipse when the lengths of the semi-major axis \(a\) and semi-minor axis \(b\) increase by given percentages.

Step 2: Know the formula for area change

The equation for the area of an ellipse is given by \(A = \pi a b\). To find the approximate change, use the formula for area: \(dA \approx \pi (b \, da + a \, db)\), where \(da\) and \(db\) are the changes in \(a\) and \(b\), respectively.

Step 3: Calculate percentage change in dimensions

Given that \(a\) increases by \(2\%\), \(da = 0.02a\). Similarly, since \(b\) increases by \(1.5\%\), \(db = 0.015b\).

Step 4: Substitute changes into formula

Replace \(da\) and \(db\) in the formula for \(dA\) from Step 2: \[dA \approx \pi (b \cdot 0.02a + a \cdot 0.015b) = \pi ab (0.02 + 0.015) = \pi ab \cdot 0.035\]

Step 5: Calculate percent change in area

The percent change in area is given by \[\text{Percent Change} = \frac{dA}{A} \times 100\% = \frac{\pi ab \cdot 0.035}{\pi ab} \times 100\% = 0.035 \times 100\% = 3.5\%\]

Key Concepts

Percentage Change

Percentage change is a way to express how much a quantity has increased or decreased in relation to its original size. In this context, we're investigating the changes in the dimensions of an ellipse's area when its major and minor axes are altered.
The change can be modeled as a percentage, providing a straightforward understanding of differences in size:

• Positive change: Indicates an increase.
• Negative change: Signifies a decrease.

For the ellipse problem, we calculated the change in each dimension:
- Semi-major axis increased by 2%, leading to a change factor for this dimension. - Semi-minor axis increased by 1.5%, also a contributing change factor. These individual changes contribute to the overall change in area.

Partial Derivatives

Partial derivatives allow us to understand how a multivariable function changes as one variable changes, while keeping others constant. For the ellipse area problem, our function is the area defined as:\[ A = \pi a b \]When solving for area changes:

• Partial Derivative with respect to \(a\): Keeps \(b\) constant and finds the rate at which the area changes with small changes in \(a\).
• Partial Derivative with respect to \(b\): Keeps \(a\) constant and finds how the area changes with small changes in \(b\).

By using partial derivatives in the formulation of the area change, the detailed effects of tiny changes in the axes can be approximated accurately. This is especially useful when variables change individually.

Differential Calculus

Differential calculus involves the study of how functions change and can be used to approximate changes in quantities. By examining an ellipse, differential calculus helps us understand small adjustments in the dimensions of the axes and their impact on the area.
In the given exercise:

• We employed differentials \(da\) and \(db\) to denote small changes in the axes.
• The formula \(dA \approx \pi (b \, da + a \, db)\) connects these differentials to the change in area.

This method offers a powerful technique to estimate variations in the ellipse's dimensions, capturing the combined growth effects precisely. It's a tool that simplifies complex changes into manageable calculations, providing a clearer mathematical picture of any variation within a system.