The radius and height of a cylinder are in the ratio 5:7 and ts volume is `550 cm^(3)`. Calculate its curved surface area in sq. cm.

Options

A. 110
B. 444
C. 220
D. 616

Answer

To solve the problem step by step, we will follow the given information about the cylinder's dimensions and its volume.

### Step 1: Understand the given ratio of radius and height
The radius (r) and height (h) of the cylinder are in the ratio of 5:7. This means we can express them in terms of a variable \( x \):
- Let \( r = 5x \)
- Let \( h = 7x \)

### Step 2: Write the formula for the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
We know that the volume is \( 550 \, \text{cm}^3 \).

### Step 3: Substitute the expressions for r and h into the volume formula
Substituting \( r \) and \( h \) into the volume formula gives:
\[ 550 = \pi (5x)^2 (7x) \]
This simplifies to:
\[ 550 = \pi (25x^2)(7x) = 175\pi x^3 \]

### Step 4: Substitute the value of \(\pi\)
Using \( \pi \approx \frac{22}{7} \):
\[ 550 = 175 \left(\frac{22}{7}\right) x^3 \]

### Step 5: Solve for \( x^3 \)
To isolate \( x^3 \), we multiply both sides by \( 7 \) and divide by \( 175 \):
\[ 550 \times 7 = 175 \times 22 x^3 \]
\[ 3850 = 3850 x^3 \]
\[ x^3 = 1 \]
Taking the cube root gives:
\[ x = 1 \]

### Step 6: Find the values of r and h
Now that we have \( x \):
- \( r = 5x = 5 \times 1 = 5 \, \text{cm} \)
- \( h = 7x = 7 \times 1 = 7 \, \text{cm} \)

### Step 7: Calculate the curved surface area (CSA)
The formula for the curved surface area of a cylinder is:
\[ \text{CSA} = 2\pi rh \]
Substituting the values of \( r \) and \( h \):
\[ \text{CSA} = 2 \times \frac{22}{7} \times 5 \times 7 \]

### Step 8: Simplify the expression
Calculating the CSA:
\[ \text{CSA} = 2 \times \frac{22}{7} \times 35 \]
\[ = \frac{2 \times 22 \times 35}{7} \]
\[ = \frac{1540}{7} = 220 \, \text{cm}^2 \]

### Final Answer
The curved surface area of the cylinder is \( 220 \, \text{cm}^2 \).

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