Answer:
To find the percent error in the volume of the cylinder when the piston compresses the gas, we first need to calculate the volume of the cylinder using the given measurements for the radius and distance.
The formula for the volume of a cylinder is:
\[ V = \pi r^2 h \]
Where:
- \( V \) is the volume of the cylinder,
- \( r \) is the radius of the circular cross-section of the piston,
- \( h \) is the distance the piston moves to compress the gas.
Given:
- Radius \( r = 3.9 \) cm with a possible error of \( \pm 0.0045 \) cm,
- Distance \( h = 3.35 \) cm with a possible error of \( \pm 0.0046 \) cm.
The maximum and minimum values for \( r \) and \( h \) are:
- Maximum radius: \( r_{\text{max}} = 3.9 + 0.0045 = 3.9045 \) cm
- Minimum radius: \( r_{\text{min}} = 3.9 - 0.0045 = 3.8955 \) cm
- Maximum distance: \( h_{\text{max}} = 3.35 + 0.0046 = 3.3546 \) cm
- Minimum distance: \( h_{\text{min}} = 3.35 - 0.0046 = 3.3454 \) cm
Using these values, we can calculate the maximum and minimum volumes and then find the percent error.
### Maximum Volume:
\[ V_{\text{max}} = \pi \times (3.9045)^2 \times 3.3546 \]
\[ V_{\text{max}} \approx \pi \times 15.2400 \times 3.3546 \]
\[ V_{\text{max}} \approx \pi \times 51.126024 \]
\[ V_{\text{max}} \approx 160.634 \, \text{cm}^3 \]
### Minimum Volume:
\[ V_{\text{min}} = \pi \times (3.8955)^2 \times 3.3454 \]
\[ V_{\text{min}} \approx \pi \times 15.1580 \times 3.3454 \]
\[ V_{\text{min}} \approx \pi \times 50.660362 \]
\[ V_{\text{min}} \approx 159.225 \, \text{cm}^3 \]
### Calculating Percent Error:
The average volume is \( V_{\text{avg}} = \frac{V_{\text{max}} + V_{\text{min}}}{2} \).
\[ V_{\text{avg}} = \frac{160.634 + 159.225}{2} \]
\[ V_{\text{avg}} = \frac{319.859}{2} \]
\[ V_{\text{avg}} = 159.9295 \, \text{cm}^3 \]
The percent error is then given by:
\[ \text{Percent Error} = \frac{|V_{\text{max}} - V_{\text{min}}|}{V_{\text{avg}}} \times 100\% \]
\[ \text{Percent Error} = \frac{|160.634 - 159.225|}{159.9295} \times 100\% \]
\[ \text{Percent Error} = \frac{1.409}{159.9295} \times 100\% \]
\[ \text{Percent Error} \approx 0.0088 \times 100\% \]
\[ \text{Percent Error} \approx 0.88\% \]
So, the percent error in the volume of the cylinder due to the uncertainties in the measurements is approximately 0.88%.