You also measure the length of a side of the cube using a ruler and get the value 3.15+0.05 cm. Calculate the volume of the cube and the uncertainty the calculated value and report it in the format V±õV, rounded appropriately. The volume of a cube is V=L³. (1m L = 1 cm3) cm3

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### Calculating the Volume and Uncertainty of a Cube

To determine the volume of a cube, you measure the length of a side using a ruler. You obtained a length of 3.15 ± 0.05 cm. The task is to calculate the volume of the cube and the uncertainty in the calculated value, and then report it in the specified format: \( V \pm \delta V \), rounded appropriately.

The formula for the volume of a cube is:
\[ V = L^3 \]
where:
- \( L \) is the length of a side of the cube.
- Given \( L = 3.15 \pm 0.05 \) cm.

1. Calculate the nominal volume:
\[ V = (3.15 \, \text{cm})^3 = 31.314375 \, \text{cm}^3 \]
After rounding, this becomes:
\[ V \approx 31.31\, \text{cm}^3 \]
   
2. Calculate the uncertainty in the volume, \( \delta V \):
\[ \delta V = |3(L \pm \delta L)^2 \cdot \delta L| \]
\[ \delta V = |3 \times (3.15 \, \text{cm})^2 \times 0.05 \, \text{cm}| \]
\[ \delta V = |3 \times 9.9225 \, \text{cm}^2 \times 0.05 \, \text{cm}| \]
\[ \delta V \approx 1.49 \, \text{cm}^3 \]

Now, expressing the calculated volume with its uncertainty in the format \( V \pm \delta V \), we get:
\[ V \pm \delta V = 31.31 \pm 1.49 \, \text{cm}^3 \]

### Interactive Problem Reporting

For recording your results:
- Enter the nominal volume in the first box.
- Enter the uncertainty in the second box.

\[ [ \text{31.31} ] \pm [ \text{1.49} ] \, \text{cm}^3 \]

Feel free to use this example as a guide in solving similar problems involving measurements and uncertainty calculations.
Transcribed Image Text:### Calculating the Volume and Uncertainty of a Cube To determine the volume of a cube, you measure the length of a side using a ruler. You obtained a length of 3.15 ± 0.05 cm. The task is to calculate the volume of the cube and the uncertainty in the calculated value, and then report it in the specified format: \( V \pm \delta V \), rounded appropriately. The formula for the volume of a cube is: \[ V = L^3 \] where: - \( L \) is the length of a side of the cube. - Given \( L = 3.15 \pm 0.05 \) cm. 1. Calculate the nominal volume: \[ V = (3.15 \, \text{cm})^3 = 31.314375 \, \text{cm}^3 \] After rounding, this becomes: \[ V \approx 31.31\, \text{cm}^3 \] 2. Calculate the uncertainty in the volume, \( \delta V \): \[ \delta V = |3(L \pm \delta L)^2 \cdot \delta L| \] \[ \delta V = |3 \times (3.15 \, \text{cm})^2 \times 0.05 \, \text{cm}| \] \[ \delta V = |3 \times 9.9225 \, \text{cm}^2 \times 0.05 \, \text{cm}| \] \[ \delta V \approx 1.49 \, \text{cm}^3 \] Now, expressing the calculated volume with its uncertainty in the format \( V \pm \delta V \), we get: \[ V \pm \delta V = 31.31 \pm 1.49 \, \text{cm}^3 \] ### Interactive Problem Reporting For recording your results: - Enter the nominal volume in the first box. - Enter the uncertainty in the second box. \[ [ \text{31.31} ] \pm [ \text{1.49} ] \, \text{cm}^3 \] Feel free to use this example as a guide in solving similar problems involving measurements and uncertainty calculations.
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What is the percent error in the accuracy in the measured volume of the cube? Use the calculated value as the expected value. Round your answer to 2 digits.

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