Question
An often-used approximation of Pi is 3.14. Calculate the percentage error in using 3.14 instead of the exact amount of Pi when calculating the area of a circle. Note: the area of a circle is A = Pir^2
Answers
GPT-4o mini
To calculate the percentage error in using 3.14 instead of the exact value of \(\pi\) when calculating the area of a circle, we can follow these steps:
1. **Formulas for the area of the circle**:
- The exact area using \(\pi\) is:
\[
A_{\text{exact}} = \pi r^2
\]
- The area using the approximation of \(\pi = 3.14\) is:
\[
A_{\text{approx}} = 3.14 r^2
\]
2. **Calculate the error in area**:
- The error in area, \(E\), can be found by taking the difference between the approximate area and the exact area:
\[
E = A_{\text{approx}} - A_{\text{exact}} = (3.14 r^2) - (\pi r^2) = (3.14 - \pi) r^2
\]
3. **Calculate the percentage error**:
- The percentage error is calculated using the formula:
\[
\text{Percentage Error} = \left(\frac{\text{Error}}{\text{Exact Value}} \right) \times 100\%
\]
- Substituting in the values we have:
\[
\text{Percentage Error} = \left(\frac{(3.14 - \pi) r^2}{\pi r^2}\right) \times 100\% = \left(\frac{3.14 - \pi}{\pi}\right) \times 100\%
\]
4. **Calculate the numerical values**:
- Using \(\pi \approx 3.14159265359\):
\[
3.14 - \pi \approx 3.14 - 3.14159265359 \approx -0.00159265359
\]
- Now calculate the percentage error:
\[
\text{Percentage Error} = \left(\frac{-0.00159265359}{\pi}\right) \times 100\%
\]
\[
\text{Percentage Error} \approx \left(\frac{-0.00159265359}{3.14159265359}\right) \times 100\% \approx -0.0507\%
\]
Thus, using \(3.14\) instead of the exact value of \(\pi\) results in a **percentage error of approximately \(-0.0507\%\)**, indicating that the approximation underestimates the area of the circle.
1. **Formulas for the area of the circle**:
- The exact area using \(\pi\) is:
\[
A_{\text{exact}} = \pi r^2
\]
- The area using the approximation of \(\pi = 3.14\) is:
\[
A_{\text{approx}} = 3.14 r^2
\]
2. **Calculate the error in area**:
- The error in area, \(E\), can be found by taking the difference between the approximate area and the exact area:
\[
E = A_{\text{approx}} - A_{\text{exact}} = (3.14 r^2) - (\pi r^2) = (3.14 - \pi) r^2
\]
3. **Calculate the percentage error**:
- The percentage error is calculated using the formula:
\[
\text{Percentage Error} = \left(\frac{\text{Error}}{\text{Exact Value}} \right) \times 100\%
\]
- Substituting in the values we have:
\[
\text{Percentage Error} = \left(\frac{(3.14 - \pi) r^2}{\pi r^2}\right) \times 100\% = \left(\frac{3.14 - \pi}{\pi}\right) \times 100\%
\]
4. **Calculate the numerical values**:
- Using \(\pi \approx 3.14159265359\):
\[
3.14 - \pi \approx 3.14 - 3.14159265359 \approx -0.00159265359
\]
- Now calculate the percentage error:
\[
\text{Percentage Error} = \left(\frac{-0.00159265359}{\pi}\right) \times 100\%
\]
\[
\text{Percentage Error} \approx \left(\frac{-0.00159265359}{3.14159265359}\right) \times 100\% \approx -0.0507\%
\]
Thus, using \(3.14\) instead of the exact value of \(\pi\) results in a **percentage error of approximately \(-0.0507\%\)**, indicating that the approximation underestimates the area of the circle.