Radius of a cylinder is equal to the side of an equilateral triangle having area `25sqrt3 cm^2` and height of the cylinder is equal to the perimeter of the triangle. Then find the volume of cylinder.

Options

A. `3000del cu.cm`
B. `3500 del cu.cm`
C. `3620 del cu. cm`
D. `2460 del cu.cm`

Explanation

To find the volume of the cylinder, we will follow these steps:

### Step 1: Find the side length of the equilateral triangle.
The area of an equilateral triangle is given by the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]
where \( a \) is the side length of the triangle. We are given that the area is \( 25\sqrt{3} \, \text{cm}^2 \).

Setting the area equal to \( 25\sqrt{3} \):
\[ \frac{\sqrt{3}}{4} a^2 = 25\sqrt{3} \]

### Step 2: Solve for \( a^2 \).
To eliminate \( \sqrt{3} \), we can multiply both sides by \( 4 \):
\[ \sqrt{3} a^2 = 100\sqrt{3} \]
Now, divide both sides by \( \sqrt{3} \):
\[ a^2 = 100 \]

### Step 3: Find \( a \).
Taking the square root of both sides:
\[ a = 10 \, \text{cm} \]

### Step 4: Determine the radius of the cylinder.
According to the problem, the radius of the cylinder is equal to the side of the equilateral triangle:
\[ \text{Radius of the cylinder} = a = 10 \, \text{cm} \]

### Step 5: Find the perimeter of the triangle.
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting the value of \( a \):
\[ P = 3 \times 10 = 30 \, \text{cm} \]

### Step 6: Determine the height of the cylinder.
According to the problem, the height of the cylinder is equal to the perimeter of the triangle:
\[ \text{Height of the cylinder} = P = 30 \, \text{cm} \]

### Step 7: Calculate the volume of the cylinder.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values of the radius \( r = 10 \, \text{cm} \) and height \( h = 30 \, \text{cm} \):
\[ V = \pi (10)^2 (30) = \pi (100)(30) = 3000\pi \, \text{cm}^3 \]

### Final Answer:
The volume of the cylinder is:
\[ \boxed{3000\pi \, \text{cm}^3} \]