If the diameter of a cylinder is `12.6 pm0.1` cm and its height is `34.2 pm0.1` , 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: Calculate the Radius
The diameter of the cylinder is given as \(12.6 \pm 0.1 \, \text{cm}\). To find the radius, we divide the diameter by 2.

\[ r = \frac{12.6}{2} = 6.3 \, \text{cm} \]

The uncertainty in the radius can be calculated as half of the uncertainty in the diameter:

\[ \Delta r = \frac{0.1}{2} = 0.05 \, \text{cm} \]

### Step 2: Write the Volume Formula
The volume \(V\) of a cylinder is given by the formula:

\[ V = \pi r^2 h \]

Where:
- \(r\) is the radius
- \(h\) is the height

### Step 3: Substitute the Values
The height of the cylinder is given as \(34.2 \pm 0.1 \, \text{cm}\). We can use the height value directly for volume calculation:

\[ h = 34.2 \, \text{cm} \]

Now substituting the values into the volume formula:

\[ V = \pi (6.3)^2 (34.2) \]

Using \(\pi \approx \frac{22}{7}\):

\[ V \approx \frac{22}{7} \times (6.3)^2 \times 34.2 \]

### Step 4: Calculate \(r^2\)
First, calculate \(r^2\):

\[ (6.3)^2 = 39.69 \]

### Step 5: Calculate the Volume
Now substitute \(r^2\) back into the volume equation:

\[ V \approx \frac{22}{7} \times 39.69 \times 34.2 \]

Calculating this step by step:

1. Calculate \(39.69 \times 34.2\):

\[ 39.69 \times 34.2 \approx 1355.658 \]

2. Now multiply by \(\frac{22}{7}\):

\[ V \approx \frac{22 \times 1355.658}{7} \approx \frac{29826.476}{7} \approx 4260.93 \, \text{cm}^3 \]

### Step 6: Round to the Nearest Significant Figure
The volume calculated is approximately \(4260.93 \, \text{cm}^3\). Rounding this to the nearest significant figure gives:

\[ V \approx 4260 \, \text{cm}^3 \]

### Final Answer
The volume of the cylinder to the nearest significant figure is:

\[ \boxed{4260 \, \text{cm}^3} \]