A hollow cylinder with outer radius 4 cm and height 2 cm is made up of I cm thick metal sheet. Whatis the volume of metal used? (Take `pi=22/7` )

Options

A. `40 cm^3`
B. `56 cm^3`
C. `44 cm^3`
D. `65 cm^3`

Answer

To find the volume of the metal used in the hollow cylinder, we can follow these steps:

### Step 1: Identify the dimensions of the hollow cylinder
- Outer radius \( R_1 = 4 \) cm
- Height \( H = 2 \) cm
- Thickness of the metal sheet = 1 cm

### Step 2: Calculate the inner radius
The inner radius \( R_2 \) can be calculated by subtracting the thickness from the outer radius:
\[ R_2 = R_1 - \text{thickness} = 4 \text{ cm} - 1 \text{ cm} = 3 \text{ cm} \]

### Step 3: Write the formula for the volume of the hollow cylinder
The volume of the metal used in the hollow cylinder can be calculated using the formula:
\[ \text{Volume of metal} = \pi H (R_1^2 - R_2^2) \]

### Step 4: Substitute the values into the formula
Using \( \pi = \frac{22}{7} \), \( H = 2 \) cm, \( R_1 = 4 \) cm, and \( R_2 = 3 \) cm, we substitute these values into the formula:
\[ \text{Volume of metal} = \frac{22}{7} \times 2 \times (4^2 - 3^2) \]

### Step 5: Calculate \( R_1^2 \) and \( R_2^2 \)
Calculate \( R_1^2 \) and \( R_2^2 \):
\[ R_1^2 = 4^2 = 16 \]
\[ R_2^2 = 3^2 = 9 \]

### Step 6: Calculate the difference \( R_1^2 - R_2^2 \)
Now, calculate the difference:
\[ R_1^2 - R_2^2 = 16 - 9 = 7 \]

### Step 7: Substitute back into the volume formula
Now substitute the difference back into the volume formula:
\[ \text{Volume of metal} = \frac{22}{7} \times 2 \times 7 \]

### Step 8: Simplify the expression
Now simplify the expression:
\[ \text{Volume of metal} = \frac{22 \times 2 \times 7}{7} \]
The \( 7 \) in the numerator and denominator cancel out:
\[ \text{Volume of metal} = 22 \times 2 = 44 \text{ cm}^3 \]

### Final Answer
The volume of metal used in the hollow cylinder is \( 44 \text{ cm}^3 \).

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