A right circular cylinder is formed. A = sum of total surface area and the area of the two bases. B = the curved surface area of this cylinder. If A : B = 3 : 2 and the volume of cylinder is 4312 `cm^(2)`, then what is the sum of area `("in" cm^(2))` of the bases of this cylinder ?

Options

A. 154
B. 308
C. 462
D. 616

Answer

To solve the problem step by step, we will follow the information given and derive the required values systematically.

### Step 1: Understand the Definitions
We have:
- \( A \) = sum of total surface area and the area of the two bases of the cylinder.
- \( B \) = curved surface area of the cylinder.

### Step 2: Formulate the Areas
The total surface area \( A \) of a right circular cylinder is given by:
\[ A = 2\pi r^2 + 2\pi rh \]
The area of the two bases is:
\[ \text{Area of two bases} = 2\pi r^2 \]
Thus, the sum \( A \) can be expressed as:
\[ A = (2\pi r^2 + 2\pi rh) + 2\pi r^2 = 4\pi r^2 + 2\pi rh \]

The curved surface area \( B \) is given by:
\[ B = 2\pi rh \]

### Step 3: Set Up the Ratio
We know from the problem statement that:
\[ \frac{A}{B} = \frac{3}{2} \]
Substituting the expressions for \( A \) and \( B \):
\[ \frac{4\pi r^2 + 2\pi rh}{2\pi rh} = \frac{3}{2} \]

### Step 4: Simplify the Ratio
Cancelling \( 2\pi rh \) from both sides:
\[ \frac{4r^2 + 2rh}{2rh} = \frac{3}{2} \]
Cross-multiplying gives:
\[ 2(4r^2 + 2rh) = 3(2rh) \]
Expanding both sides:
\[ 8r^2 + 4rh = 6rh \]
Rearranging gives:
\[ 8r^2 = 6rh - 4rh \]
\[ 8r^2 = 2rh \]
Dividing both sides by \( 2r \) (assuming \( r \neq 0 \)):
\[ 4r = h \quad (1) \]

### Step 5: Use the Volume Formula
The volume \( V \) of the cylinder is given by:
\[ V = \pi r^2 h \]
Given \( V = 4312 \, \text{cm}^3 \), substituting \( h \) from equation (1):
\[ 4312 = \pi r^2 (4r) = 4\pi r^3 \]
Thus:
\[ r^3 = \frac{4312}{4\pi} \]

### Step 6: Calculate \( r \)
Using \( \pi \approx \frac{22}{7} \):
\[ r^3 = \frac{4312 \times 7}{4 \times 22} = \frac{30184}{88} = 343 \]
Taking the cube root:
\[ r = \sqrt[3]{343} = 7 \, \text{cm} \]

### Step 7: Find \( h \)
Using equation (1):
\[ h = 4r = 4 \times 7 = 28 \, \text{cm} \]

### Step 8: Calculate the Area of the Bases
The area of the bases of the cylinder is given by:
\[ \text{Area of two bases} = 2\pi r^2 = 2\pi (7^2) = 2\pi \times 49 = 98\pi \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area} = 98 \times \frac{22}{7} = 308 \, \text{cm}^2 \]

### Final Answer
The sum of the area of the bases of the cylinder is:
\[ \boxed{308 \, \text{cm}^2} \]