A solid cylinder has total surface area of 462 sq. cm. Its curved surface area is `(1)/(3)`rd of the total surface area. Then the radius of the cylinder is

Options

A. 7 cm
B. 3.5 cm
C. 9 cm
D. 11 cm

Answer

To solve the problem, let's follow these steps:

### Step 1: Understand the given information
We know that the total surface area (TSA) of the cylinder is 462 sq. cm and the curved surface area (CSA) is one-third of the total surface area.

### Step 2: Calculate the curved surface area
Since the curved surface area is one-third of the total surface area:
\[ \text{CSA} = \frac{1}{3} \times \text{TSA} = \frac{1}{3} \times 462 = 154 \text{ sq. cm} \]

### Step 3: Write the formula for total surface area
The total surface area of a cylinder is given by the formula:
\[ \text{TSA} = 2\pi rh + 2\pi r^2 \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 4: Substitute the known values into the formula
We can substitute the total surface area and the curved surface area into the formula:
\[ 462 = 2\pi rh + 2\pi r^2 \]
We already know that:
\[ \text{CSA} = 2\pi rh = 154 \]
So we can replace \( 2\pi rh \) in the total surface area formula:
\[ 462 = 154 + 2\pi r^2 \]

### Step 5: Solve for \( 2\pi r^2 \)
Now, isolate \( 2\pi r^2 \):
\[ 2\pi r^2 = 462 - 154 = 308 \]

### Step 6: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ 2 \times \frac{22}{7} \times r^2 = 308 \]

### Step 7: Simplify the equation
Multiply both sides by 7 to eliminate the fraction:
\[ 44r^2 = 308 \times 7 \]
Calculate \( 308 \times 7 \):
\[ 308 \times 7 = 2156 \]
So we have:
\[ 44r^2 = 2156 \]

### Step 8: Solve for \( r^2 \)
Now, divide both sides by 44:
\[ r^2 = \frac{2156}{44} = 49 \]

### Step 9: Find the radius \( r \)
Taking the square root of both sides gives:
\[ r = \sqrt{49} = 7 \text{ cm} \]

### Conclusion
The radius of the cylinder is \( 7 \) cm.

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