Problem 37 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The area of the ellipse has approximately increased by 3.53% when the major axis length increases by 2% and the minor axis length increases by 1.5%.

Step 1: Determine the new lengths of the axes after the increase

First, let us find the new lengths of the axes after the increase in percentage. To do so, we multiply each axis length by \((1 + \textit{percentage increase})\). New length of \(2a\): \((1+0.02)(2a) = 2a(1.02)\) New length of \(2b\): \((1+0.015)(2b) = 2b(1.015)\)

Step 2: Calculate the new area of the ellipse after the increase

Next, we use the new axis lengths to calculate the new area of the ellipse. To do this, we use the ellipse area formula, \(A = \pi a b\), and substitute the new axis lengths. New area, \(A_{new} = \pi (a(1.02))(b(1.015)) = \pi (1.02)(1.015)ab\)

Step 3: Calculate the percentage change in the area

We will now calculate the percentage change in the area of the ellipse. To do this, we use the percent change formula: Percent change = \(\frac{\text{New area - Old area}}{\text{Old area}} \times 100\) \(\text{Percent change} = \frac{A_{new} - A}{A} \times 100\) Substitute the values of \(A\) and \(A_{new}\): \(\text{Percent change} = \frac{\pi (1.02)(1.015)ab - \pi ab}{\pi ab} \times 100\) Factor out \(\pi ab\) from both terms in the numerator: \(\text{Percent change} = \frac{\pi ab((1.02)(1.015) - 1)}{\pi ab} \times 100\) Cancel out the \(\pi ab\) terms: \(\text{Percent change} = ((1.02)(1.015) - 1) \times 100\)

Step 4: Calculate the numerical value of the percent change

Finally, compute the numerical value of the percent change using the expression we derived: \(\text{Percent change} = ((1.02)(1.015) - 1) \times 100 \approx 3.53 \%\) The area of the ellipse has approximately increased by \(3.53\%\) when \(a\) increases by \(2\%\) and \(b\) increases by \(1.5\%\).

Key Concepts

Understanding Percent Change

Percent change measures how much a quantity has increased or decreased in relation to its original value. It's commonly used to compare the initial state to a changed state. To calculate percent change, use the formula:\[ \text{Percent change} = \frac{\text{New value} - \text{Old value}}{\text{Old value}} \times 100 \]This formula helps in determining the relative increase or decrease in a value.

• If the result is positive, it indicates a percentage increase.
• If the result is negative, it shows a percentage decrease.

In the context of our ellipse problem, we apply this formula to find out how the area changes with variations in the axis lengths.

Exploring Axis Lengths in an Ellipse

An ellipse is defined by its two axes:

• The major axis, stretching across the widest part of the ellipse.
• The minor axis, crossing the shortest section.

For an ellipse with axes lengths of \(2a\) and \(2b\), these define the horizontal and vertical spreads from the center to the boundary.When calculating the area of an ellipse, these axes play a crucial role. The standard formula is:\[ A = \pi a b \]Where \(a\) and \(b\) represent the semi-major and semi-minor axes lengths.In our problem, slight increases in these lengths due to percent changes in \(a\) and \(b\) lead to a new area, reflecting how the ellipse broadens as these axes grow.

Calculating Percentage Increase

Percentage increase is a specific type of percent change, focusing on how much a quantity has grown compared to its original measure. To find the percentage increase, one might calculate:\[ \text{Percentage increase} = \frac{\text{New value - Old value}}{\text{Old value}} \times 100 \]This formula highlights the growth in a quantity when additional attributes are applied. In our ellipse problem:

• Axis \(a\) increases by 2%.
• Axis \(b\) grows by 1.5%.

These specific increases affect the overall area. By recalculating the area with updated axes values, the percentage increase in the area can be derived, showing us the degree of expansion that has occurred. The result, in this case, turns out to be approximately 3.53%, showing how even small changes in the primary dimensions can significantly impact the overall shape.