If the diameter of a cylinder is `12.6 pm0.1` cm and its height is `34.2 pm0.1`cm , then find the volume of the cylinder to the nearest significant figure .

Options

A. `426.4pm81 cm^3`
B. `4260pm81cm^3`
C. `4300pm80cm^3`
D. `4260pm80cm^3`

Answer

To find the volume of the cylinder given its diameter and height, we will follow these steps:

### Step 1: Identify the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

### Step 2: Convert diameter to radius
The diameter \( d \) of the cylinder is given as \( 12.6 \pm 0.1 \) cm. The radius \( r \) is half of the diameter:
\[ r = \frac{d}{2} = \frac{12.6}{2} = 6.3 \text{ cm} \]

### Step 3: Substitute the values into the volume formula
Now, we substitute the radius and height into the volume formula. The height \( h \) is given as \( 34.2 \pm 0.1 \) cm.
\[ V = \pi (6.3)^2 (34.2) \]

### Step 4: Calculate the volume
First, calculate \( (6.3)^2 \):
\[ (6.3)^2 = 39.69 \]
Now substitute this value into the volume formula:
\[ V = \pi \times 39.69 \times 34.2 \]
Calculating this gives:
\[ V \approx 3.14159 \times 39.69 \times 34.2 \approx 4262.22 \text{ cm}^3 \]

### Step 5: Calculate the uncertainty in volume
To find the uncertainty in volume, we need to use the formula for propagation of uncertainty. The relative uncertainty in volume \( \frac{\Delta V}{V} \) can be calculated using:
\[ \frac{\Delta V}{V} = 2 \frac{\Delta d}{d} + \frac{\Delta h}{h} \]
where \( \Delta d = 0.1 \) cm and \( \Delta h = 0.1 \) cm.

Calculating each term:
- The diameter \( d = 12.6 \) cm, so:
\[ \frac{\Delta d}{d} = \frac{0.1}{12.6} \approx 0.0079365 \]
- The height \( h = 34.2 \) cm, so:
\[ \frac{\Delta h}{h} = \frac{0.1}{34.2} \approx 0.00292398 \]

Now substituting these values into the relative uncertainty formula:
\[ \frac{\Delta V}{V} = 2 \times 0.0079365 + 0.00292398 \approx 0.018796 \]

### Step 6: Calculate the absolute uncertainty in volume
Now, calculate \( \Delta V \):
\[ \Delta V = V \times \frac{\Delta V}{V} = 4262.22 \times 0.018796 \approx 80.10 \text{ cm}^3 \]

### Step 7: Write the final result with uncertainty
Thus, the volume of the cylinder can be expressed as:
\[ V = 4262.22 \pm 80.10 \text{ cm}^3 \]

### Step 8: Round to the nearest significant figure
Since the volume is \( 4262.22 \), we round it to the nearest significant figure based on the uncertainty \( 80.10 \), which affects the hundreds place. Therefore, we round to:
\[ V \approx 4260 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder to the nearest significant figure is:
\[ \boxed{4260 \text{ cm}^3} \]