A cylinder 84 cm long is made of steel. Its external and internal diameters are 10 cm and 8 cm respectively. What is the volume of the steel in the cylinder (in `10^(-3) m^3` and correct up to three decimalplaces) ?

Options

A. 2.112
B. 2.376
C. 4.752
D. 9.504

Answer

To find the volume of steel in the hollow cylinder, we will follow these steps:

### Step 1: Identify the dimensions of the cylinder
- Length (height) of the cylinder, \( h = 84 \) cm
- External diameter, \( D = 10 \) cm
- Internal diameter, \( d = 8 \) cm

### Step 2: Calculate the external and internal radii
- External radius, \( R = \frac{D}{2} = \frac{10}{2} = 5 \) cm
- Internal radius, \( r = \frac{d}{2} = \frac{8}{2} = 4 \) cm

### Step 3: Use the formula for the volume of a hollow cylinder
The volume \( V \) of a hollow cylinder is given by the formula:
\[ V = \pi h (R^2 - r^2) \]
where:
- \( R \) is the external radius,
- \( r \) is the internal radius,
- \( h \) is the height of the cylinder.

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:
\[ V = \pi \times 84 \times (5^2 - 4^2) \]
Calculating \( 5^2 \) and \( 4^2 \):
\[ 5^2 = 25 \quad \text{and} \quad 4^2 = 16 \]
Thus,
\[ R^2 - r^2 = 25 - 16 = 9 \]
Now substituting this back into the volume formula:
\[ V = \pi \times 84 \times 9 \]

### Step 5: Calculate the volume
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 84 \times 9 \]
Calculating \( 84 \times 9 = 756 \):
\[ V = \frac{22}{7} \times 756 \]
Now, calculate \( \frac{756}{7} = 108 \):
\[ V = 22 \times 108 = 2376 \text{ cm}^3 \]

### Step 6: Convert the volume to cubic meters
To convert cubic centimeters to cubic meters:
\[ 1 \text{ cm}^3 = 10^{-6} \text{ m}^3 \]
Thus,
\[ V = 2376 \times 10^{-6} \text{ m}^3 = 2.376 \times 10^{-3} \text{ m}^3 \]

### Step 7: Round to three decimal places
The final answer, rounded to three decimal places, is:
\[ \boxed{2.376 \times 10^{-3} \text{ m}^3} \]