Ratio between curved surface area and total surface area of a circular cylinder is 3 : 5. If curved surface area is `7392 cm^3` then what is the height of cylinder.

Options

A. 48
B. 21
C. 47
D. 42

Explanation

To solve the problem step by step, we will follow the given information and formulas related to the curved surface area (CSA) and total surface area (TSA) of a circular cylinder.

### Step 1: Understand the given ratio
We know that the ratio of the curved surface area (CSA) to the total surface area (TSA) of a circular cylinder is given as 3:5.

### Step 2: Write the formulas for CSA and TSA
The formulas for the curved surface area and total surface area of a cylinder are:
- Curved Surface Area (CSA) = \(2 \pi r h\)
- Total Surface Area (TSA) = \(2 \pi r (r + h)\)

### Step 3: Set up the equation based on the ratio
From the ratio \( \frac{CSA}{TSA} = \frac{3}{5} \), we can set up the equation:
\[ \frac{2 \pi r h}{2 \pi r (r + h)} = \frac{3}{5} \]
This simplifies to:
\[ \frac{h}{r + h} = \frac{3}{5} \]

### Step 4: Cross-multiply to solve for h and r
Cross-multiplying gives us:
\[ 5h = 3(r + h) \]
Expanding this, we get:
\[ 5h = 3r + 3h \]
Rearranging the equation:
\[ 5h - 3h = 3r \implies 2h = 3r \]
Thus, we have:
\[ h = \frac{3}{2}r \]

### Step 5: Use the given CSA to find r and h
We know that the curved surface area (CSA) is given as \(7392 \, \text{cm}^2\):
\[ 2 \pi r h = 7392 \]
Substituting \(h = \frac{3}{2}r\) into the CSA formula:
\[ 2 \pi r \left(\frac{3}{2}r\right) = 7392 \]
This simplifies to:
\[ 3 \pi r^2 = 7392 \]

### Step 6: Solve for r
Now, we can solve for \(r^2\):
\[ r^2 = \frac{7392}{3 \pi} \]
Using \(\pi \approx \frac{22}{7}\):
\[ r^2 = \frac{7392 \times 7}{3 \times 22} = \frac{51744}{66} = 784 \]
Taking the square root gives:
\[ r = \sqrt{784} = 28 \, \text{cm} \]

### Step 7: Find the height h
Now, substitute \(r\) back to find \(h\):
\[ h = \frac{3}{2}r = \frac{3}{2} \times 28 = 42 \, \text{cm} \]

### Final Answer
The height of the cylinder is \(42 \, \text{cm}\).

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