The ratio of curved surface area of a right circular cylinder to the total area of its two bases is 2 : 1. If the total surface area of cylinder is 23100 `cm^2` , then what is the volume (in `cm^3` ) of cylinder?

Options

A. 247200
B. 269500
C. 312500
D. 341800

Answer

To solve the problem step by step, we will follow the given information and derive the necessary values to find the volume of the cylinder.

### Step 1: Understand the given information
We are given that the ratio of the curved surface area (CSA) of a right circular cylinder to the total area of its two bases is 2:1. We also know the total surface area (TSA) of the cylinder is 23100 cm².

### Step 2: Define the formulas
1. The curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi rh \]
2. The area of the two bases is:
\[ \text{Area of two bases} = 2\pi r^2 \]
3. The total surface area (TSA) of the cylinder is:
\[ \text{TSA} = \text{CSA} + \text{Area of two bases} = 2\pi rh + 2\pi r^2 \]

### Step 3: Set up the ratio
From the problem, we have:
\[ \frac{\text{CSA}}{\text{Area of two bases}} = \frac{2}{1} \]
Substituting the formulas:
\[ \frac{2\pi rh}{2\pi r^2} = \frac{2}{1} \]
This simplifies to:
\[ \frac{h}{r} = 2 \implies h = 2r \]

### Step 4: Substitute h in TSA
Now we substitute \( h = 2r \) into the TSA formula:
\[ \text{TSA} = 2\pi r(2r) + 2\pi r^2 = 4\pi r^2 + 2\pi r^2 = 6\pi r^2 \]
We know the TSA is 23100 cm²:
\[ 6\pi r^2 = 23100 \]

### Step 5: Solve for r²
Now, we solve for \( r^2 \):
\[ r^2 = \frac{23100}{6\pi} \]
Using \( \pi \approx 3.14 \):
\[ r^2 = \frac{23100}{6 \times 3.14} = \frac{23100}{18.84} \approx 1225 \]

### Step 6: Find r
Taking the square root of both sides:
\[ r = \sqrt{1225} = 35 \text{ cm} \]

### Step 7: Find h
Using \( h = 2r \):
\[ h = 2 \times 35 = 70 \text{ cm} \]

### Step 8: Calculate the volume
The volume \( V \) of the cylinder is given by:
\[ V = \pi r^2 h \]
Substituting the values:
\[ V = \pi (35^2)(70) = \pi (1225)(70) = 85750\pi \]
Using \( \pi \approx 3.14 \):
\[ V \approx 85750 \times 3.14 \approx 269,495 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is approximately 269500 cm³.