The volumes of a solid right circular cylinder is `5236 cm^(3)` and its height is 34cm. What is its curved surface area ( in `cm^(2)` ) ? (Take `pi = underset( 7 ) overset( 22)` )

Options

A. 1804
B. 1496
C. 1573
D. 1650

Answer

To find the curved surface area of a solid right circular cylinder given its volume and height, we can follow these steps:

### Step 1: Understand the formulas
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder. The curved surface area \( A \) of a cylinder is given by:
\[ A = 2\pi rh \]

### Step 2: Substitute the known values
We know:
- Volume \( V = 5236 \, \text{cm}^3 \)
- Height \( h = 34 \, \text{cm} \)
- \( \pi = \frac{22}{7} \)

Substituting the known values into the volume formula:
\[ 5236 = \frac{22}{7} r^2 \times 34 \]

### Step 3: Rearrange to find \( r^2 \)
To isolate \( r^2 \), we can rearrange the equation:
\[ r^2 = \frac{5236 \times 7}{22 \times 34} \]

### Step 4: Calculate \( r^2 \)
Calculating the right-hand side:
- First, calculate \( 22 \times 34 = 748 \)
- Then, calculate \( 5236 \times 7 = 36652 \)

Now substitute these values:
\[ r^2 = \frac{36652}{748} \]

### Step 5: Simplify \( r^2 \)
Now we can simplify \( r^2 \):
\[ r^2 = 49 \]
Taking the square root gives:
\[ r = 7 \, \text{cm} \]

### Step 6: Calculate the curved surface area
Now that we have the radius, we can find the curved surface area using the formula:
\[ A = 2\pi rh \]
Substituting the values:
\[ A = 2 \times \frac{22}{7} \times 7 \times 34 \]

### Step 7: Simplify the expression
The \( 7 \) in the numerator and denominator cancels out:
\[ A = 2 \times 22 \times 34 \]
Calculating this:
\[ A = 44 \times 34 = 1496 \, \text{cm}^2 \]

### Final Answer
The curved surface area of the cylinder is \( 1496 \, \text{cm}^2 \).

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