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Chapter 3: Problem 41

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

Short Answer

Expert verified
a. 0.24 m虏 b. 1.91%

Step by step solution

01

Calculate the Original Area

Calculate the original area of the circle using the formula for the area of a circle: \( A = \pi r^2 \). The original radius is 2.00 m. \[ A_{\text{original}} = \pi (2.00)^2 = 4\pi \approx 12.57 \, \text{m}^2 \]
02

Calculate the New Area

Calculate the new area of the circle using the increased radius of 2.02 m with the same formula for the area of a circle. \[ A_{\text{new}} = \pi (2.02)^2 = \pi (4.0804) \approx 12.81 \, \text{m}^2 \]
03

Determine the Change in Area

Subtract the original area from the new area to find the change in area. \[ \Delta A = A_{\text{new}} - A_{\text{original}} = 12.81 - 12.57 = 0.24 \, \text{m}^2 \]
04

Calculate the Percentage Change

Express the change in area as a percentage of the original area. Use the formula: \[ \text{percentage change} = \left( \frac{\Delta A}{A_{\text{original}}} \right) \times 100\%% \]Substitute the known values:\[ \text{percentage change} = \left( \frac{0.24}{12.57} \right) \times 100\% \approx 1.91\% \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circle Area Calculation
Understanding how to calculate the area of a circle is crucial for various applications in both mathematics and real-world scenarios. The formula for the area of a circle is given by: \( A = \pi r^2 \), where \( A \) represents the area and \( r \) is the radius.
  • The symbol \( \pi \) (pi) is a mathematical constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter.
  • To compute the area, you simply square the radius (multiply the radius by itself) and then multiply the result by \( \pi \).
For example, if the radius of a circle is 2.00 m, the calculation would be: \[ A = \pi (2.00)^2 = 4\pi \] Approximately, this area equals 12.57 square meters. Converting radius measurements to area can help in practical fields like design, architecture, and more.
Radius Increase
Increasing a circle's radius affects its area, and understanding this relationship is key in calculus problem-solving. The radius is a linear measurement from the center of the circle to any point on its boundary.
  • Small changes in the radius may lead to significant changes in the area, showing how sensitive the area is to radius alterations.
  • For instance, with an original radius of 2.00 m, if it increases to 2.02 m, even this small change will create a noticeable difference in the total area.
In the given example, with the radius increase to 2.02 m, the new area calculation is:\[ A_{\text{new}} = \pi (2.02)^2 = 12.81 \, \text{m}^2 \]This demonstrates the exponential impact of squaring the radius on the total area, making even minor radius increases influential.
Percentage Change Calculation
Calculating percentage change is a useful analytic tool to understand how one quantity changes relative to another. It involves comparing a change against an original value. The formula for percentage change is:\[ \text{percentage change} = \left( \frac{\text{change}}{\text{original value}} \right) \times 100\%% \]
  • In the context of circles, first determine the change in area as the difference between the new and original areas.
  • Then compare this change to the original area to express it as a percentage.
In the circle example, the change in area was 0.24 m虏, and the original area was approximately 12.57 m虏. Hence, the percentage change is:\[ \left( \frac{0.24}{12.57} \right) \times 100\% \approx 1.91\% \]This percentage helps quantify the impact of the radius increase, making the analysis clearer and easier to communicate.

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