Total surface area of a right circular cylinder is 1848 cm . The ratio of its total surface area to the curved surface area is 3: 1. The volume of the cylinder is:(Take `pi = (22)/(7) `)

Options

A. 3696 `cm^(3)`
B. `4312 cm^(3)`
C. `4002 cm^(3)`
D. `4851 cm^(2)`

Answer

To solve the problem step by step, we will use the formulas for the total surface area and the curved surface area of a right circular cylinder.

### Step 1: Understand the formulas
The total surface area (TSA) of a right circular cylinder is given by:
\[ \text{TSA} = 2\pi r(h + r) \]
The curved surface area (CSA) of a right circular cylinder is given by:
\[ \text{CSA} = 2\pi rh \]

### Step 2: Set up the ratio
We are given that the ratio of the total surface area to the curved surface area is 3:1. This can be expressed as:
\[ \frac{\text{TSA}}{\text{CSA}} = \frac{3}{1} \]
Substituting the formulas, we have:
\[ \frac{2\pi r(h + r)}{2\pi rh} = \frac{3}{1} \]
This simplifies to:
\[ \frac{h + r}{h} = 3 \]

### Step 3: Solve for \(h\) in terms of \(r\)
From the equation \(\frac{h + r}{h} = 3\), we can cross-multiply:
\[ h + r = 3h \]
Rearranging gives us:
\[ 2h = r \quad \Rightarrow \quad r = 2h \]

### Step 4: Substitute into the total surface area
We know the total surface area is given as 1848 cm². Substituting \(r = 2h\) into the total surface area formula:
\[ \text{TSA} = 2\pi (2h)(h + 2h) = 2\pi (2h)(3h) = 12\pi h^2 \]
Setting this equal to 1848 cm²:
\[ 12\pi h^2 = 1848 \]

### Step 5: Solve for \(h^2\)
Substituting \(\pi = \frac{22}{7}\):
\[ 12 \times \frac{22}{7} h^2 = 1848 \]
Multiplying both sides by 7:
\[ 12 \times 22 h^2 = 1848 \times 7 \]
Calculating \(1848 \times 7\):
\[ 1848 \times 7 = 12936 \]
Now we have:
\[ 264h^2 = 12936 \]
Dividing both sides by 264:
\[ h^2 = \frac{12936}{264} = 49 \]
Taking the square root:
\[ h = 7 \text{ cm} \]

### Step 6: Find \(r\)
Using \(r = 2h\):
\[ r = 2 \times 7 = 14 \text{ cm} \]

### Step 7: Calculate the volume of the cylinder
The volume \(V\) of the cylinder is given by:
\[ V = \pi r^2 h \]
Substituting the values of \(r\) and \(h\):
\[ V = \frac{22}{7} \times (14)^2 \times 7 \]
Calculating \(14^2\):
\[ 14^2 = 196 \]
Now substituting:
\[ V = \frac{22}{7} \times 196 \times 7 \]
The \(7\) cancels out:
\[ V = 22 \times 196 = 4312 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is:
\[ \boxed{4312 \text{ cm}^3} \]