The curved surface area and the total surface area of a cylinder are in the ratio 1:2. If the total surface area of the right cylinder is `616 cm^(2)`, then its volume is :

Options

A. `1232 cm^(3)`
B. `1848 cm^(3)`
C. `1632 cm^(3)`
D. `1078 cm^(3)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the curved surface area (CSA) and total surface area (TSA) of the cylinder.
The problem states that the ratio of the curved surface area to the total surface area is 1:2.

### Step 2: Write the formulas for CSA and TSA of a cylinder.
- The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]
- The formula for the total surface area (TSA) of a cylinder is given by:
\[ \text{TSA} = 2\pi r(h + r) \]

### Step 3: Set up the equation based on the ratio.
From the ratio given (CSA : TSA = 1 : 2), we can express this as:
\[ \frac{CSA}{TSA} = \frac{1}{2} \]
Substituting the formulas, we have:
\[ \frac{2\pi rh}{2\pi r(h + r)} = \frac{1}{2} \]
This simplifies to:
\[ \frac{h}{h + r} = \frac{1}{2} \]

### Step 4: Cross-multiply to solve for h and r.
Cross-multiplying gives:
\[ 2h = h + r \]
Rearranging this, we find:
\[ h = r \]

### Step 5: Use the total surface area to find r and h.
We know the total surface area (TSA) is given as 616 cm². Using the TSA formula:
\[ TSA = 2\pi r(h + r) \]
Substituting \(h = r\):
\[ 616 = 2\pi r(2r) \]
This simplifies to:
\[ 616 = 4\pi r^2 \]

### Step 6: Solve for r.
Now, substituting \(\pi \approx \frac{22}{7}\):
\[ 616 = 4 \times \frac{22}{7} \times r^2 \]
Multiplying both sides by 7:
\[ 616 \times 7 = 88r^2 \]
Calculating \(616 \times 7\):
\[ 4312 = 88r^2 \]
Dividing both sides by 88:
\[ r^2 = \frac{4312}{88} \]
Calculating this gives:
\[ r^2 = 49 \]
Taking the square root:
\[ r = 7 \text{ cm} \]
Since \(h = r\), we also have:
\[ h = 7 \text{ cm} \]

### Step 7: Calculate the volume of the cylinder.
The volume \(V\) of a cylinder is given by:
\[ V = \pi r^2 h \]
Substituting the values:
\[ V = \frac{22}{7} \times 7^2 \times 7 \]
Calculating:
\[ V = \frac{22}{7} \times 49 \times 7 \]
The \(7\) in the denominator cancels out:
\[ V = 22 \times 49 \]
Calculating \(22 \times 49\):
\[ V = 1078 \text{ cm}^3 \]

### Final Answer:
The volume of the cylinder is \(1078 \text{ cm}^3\).

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