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Step-by-Step Solution
Step 1: Write down the known measurements
The diameter (d) of the cylinder is measured as 12.6 ± 0.1 cm, and the height (h) is measured as 34.2 ± 0.1 cm.
Step 2: Express the volume formula
The volume of a cylinder is given by the formula:
$V = \pi r^2 h$
Since $r = \frac{d}{2}$, the formula becomes:
$V = \pi \frac{d^2}{4} h$
Step 3: Substitute the measured values
Substitute $d = 12.6\text{ cm}$ and $h = 34.2\text{ cm}$, and take $\pi \approx 3.14$:
$V = 3.14 \times \frac{(12.6)^2}{4} \times 34.2$
Carrying out the calculation yields:
$V \approx 4260 \text{ cm}^3
$
Step 4: Calculate the relative error in volume
The relative (fractional) error in volume $ \frac{\Delta V}{V} $ is the sum of the fractional errors appropriately accounted for the squared quantity of diameter and the height:
$\frac{\Delta V}{V} = 2 \cdot \frac{\Delta d}{d} + \frac{\Delta h}{h}$
Here, $\Delta d = 0.1\text{ cm}$, $d = 12.6\text{ cm}$, $ \Delta h = 0.1\text{ cm}$, and $h = 34.2\text{ cm}.$ Thus,
$\frac{\Delta V}{V} = 2 \left(\frac{0.1}{12.6}\right) + \frac{0.1}{34.2} \approx 0.0188
$
Step 5: Find the absolute error in the volume
Once the fractional error is known, the absolute error in the volume ($\Delta V$) is:
$\Delta V = \left(\frac{\Delta V}{V}\right) \times V = 0.0188 \times 4260 \approx 80 \text{ cm}^3
$
Step 6: State the final result with significant figures
The volume of the cylinder, including the error, is:
$ (4260 \pm 80) \text{ cm}^3
