Problem 51 - University Calculus: Early Transcendentals

Question

The radius of a circle is increased from 2.00 to \(2.02 \mathrm{m}\) a. Estimate the resulting change in area. b. Express the estimate as a percentage of the circle's original area.

Step-by-Step Solution

Answer

The change in area is \(0.0804\pi \mathrm{m^2}\) or about 2.01% of the original area.

Step 1: Calculate the original area

The formula for the area of a circle is \( A = \pi r^2 \). Given that the original radius \( r \) is 2.00 m, substitute into the formula to find the area: \[A = \pi (2.00)^2 = 4\pi \mathrm{m^2}.\]

Step 2: Calculate the new area

Now, calculate the new area with the increased radius of 2.02 m. Use the same formula for the area: \[A' = \pi (2.02)^2 = 4.0804\pi \mathrm{m^2}.\]

Step 3: Estimate the change in area

The change in area \( \Delta A \) is the difference between the new area and the original area:\[\Delta A = A' - A = 4.0804\pi - 4\pi = 0.0804\pi \mathrm{m^2}.\]

Step 4: Calculate percentage change in area

To express the change in area as a percentage, use the formula:\[\text{Percentage Change} = \left(\frac{\Delta A}{A}\right) \times 100\%\]Substitute \( \Delta A = 0.0804\pi \mathrm{m^2} \) and \( A = 4\pi \mathrm{m^2} \):\[\text{Percentage Change} = \left(\frac{0.0804\pi}{4\pi}\right) \times 100\% = 2.01\%\]

Key Concepts

Understanding Radius Increase

When we talk about a 'radius increase,' we are simply discussing how much longer the radius becomes in a circle. A circle's radius is the distance from its center to any point on its edge. If you start with a radius of 2 meters and increase it to 2.02 meters, the radius has increased by 0.02 meters.
This might seem like a tiny change, but because the circle's area is proportional to the square of the radius, even a small increase in the radius can make a noticeable difference in the area.
Whenever the radius is increased, you should think about how this affects the circle's dimensions and how it might impact calculations like area or circumference.

Calculating Circle Area

The area of a circle is calculated using the formula: \[ A = \pi r^2 \] where \( A \) represents the area and \( r \) is the radius of the circle. This formula shows that as the radius changes, the area changes in proportion to the square of that radius.

Let's consider the given example:

• Original radius: 2.00 meters
• New radius: 2.02 meters

First, calculate the original area using the formula: \[ A = \pi (2.00)^2 = 4\pi \text{ m}^2 \] Next, calculate the area with the new, slightly larger radius: \[ A' = \pi (2.02)^2 = 4.0804\pi \text{ m}^2 \] You can see that the area slightly increases due to the squared term in the formula. Small changes in the radius lead to corresponding changes in the area. This exercise should give you a strong grasp of how radius changes affect area.

Determining Percentage Change

Once you know the change in area, expressing it as a percentage can help you understand the magnitude of that change relative to the original area. Percentage change tells you how much the circle's area has increased or decreased in comparison to its initial size.

Here's how you calculate it:

• First, determine the change in area, \( \Delta A \), which is the difference between the new area and the original area: \[ \Delta A = A' - A = 4.0804\pi - 4\pi = 0.0804\pi \text{ m}^2 \]
• Then, use the formula for percentage change: \[\text{Percentage Change} = \left( \frac{\Delta A}{A} \right) \times 100\% \] Substitute the known values: \[\text{Percentage Change} = \left( \frac{0.0804\pi}{4\pi} \right) \times 100\% = 2.01\% \]

This percentage change tells us that the area of the circle has increased by approximately 2.01% due to the increase in radius. It highlights how small adjustments in the radius can lead to significant changes in the area, especially when expressed in percentage terms.