If the volume of the cylinder is 3564 `cm^3`. The external radius of the cylinder is 12 cm and thickness is 3 cm. Then find the length of the cylinder.

Options

A. 14 cm
B. 18 cm
C. 16 cm
D. 10 cm

Answer

To solve the problem step by step, we need to find the length of the cylinder given its volume, external radius, and thickness.

### Step 1: Identify the given values
- Volume of the cylinder (V) = 3564 cm³
- External radius (R) = 12 cm
- Thickness of the cylinder = 3 cm

### Step 2: Calculate the internal radius
The internal radius (r) can be calculated by subtracting the thickness from the external radius:
\[ r = R - \text{thickness} = 12 \, \text{cm} - 3 \, \text{cm} = 9 \, \text{cm} \]

### Step 3: Write the formula for the volume of a hollow cylinder
The volume of a hollow cylinder is given by the formula:
\[ V = \pi h (R^2 - r^2) \]
Where:
- \(V\) = Volume
- \(h\) = Height (or length of the cylinder)
- \(R\) = External radius
- \(r\) = Internal radius

### Step 4: Substitute the known values into the formula
Substituting the values we have:
\[ 3564 = \frac{22}{7} \cdot h \cdot (12^2 - 9^2) \]
Calculating \(12^2\) and \(9^2\):
\[ 12^2 = 144 \quad \text{and} \quad 9^2 = 81 \]
Thus,
\[ R^2 - r^2 = 144 - 81 = 63 \]
Now substituting back into the volume equation:
\[ 3564 = \frac{22}{7} \cdot h \cdot 63 \]

### Step 5: Simplify the equation
To simplify, we can multiply both sides by \(7\) to eliminate the fraction:
\[ 3564 \cdot 7 = 22 \cdot h \cdot 63 \]
Calculating \(3564 \cdot 7\):
\[ 3564 \cdot 7 = 24948 \]
So, we have:
\[ 24948 = 22 \cdot h \cdot 63 \]

### Step 6: Divide both sides by \(22 \cdot 63\)
Calculating \(22 \cdot 63\):
\[ 22 \cdot 63 = 1386 \]
Now, divide both sides by \(1386\):
\[ h = \frac{24948}{1386} \]

### Step 7: Calculate the height (length of the cylinder)
Now, performing the division:
\[ h = 18 \, \text{cm} \]

### Final Answer
The length of the cylinder is **18 cm**.

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