A conical vessel has radius 4 cm, and its curved surface area is `20pi cm^2` . Find the volume of conical vessel?

Options

A. `16 pi cm^3`
B. ` 14 pi cm^3`
C. `26 pi cm^3`
D. `64pi//3 cm^3`

Answer

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

### Step 1: Identify the given values
We are given:
- Radius (r) = 4 cm
- Curved Surface Area (CSA) = 20π cm²

### Step 2: Use the formula for the curved surface area of a cone
The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r L \]
where L is the slant height of the cone.

### Step 3: Substitute the known values into the CSA formula
Substituting the given values into the CSA formula:
\[ 20\pi = \pi \times 4 \times L \]

### Step 4: Simplify the equation to find L
Dividing both sides by π:
\[ 20 = 4L \]
Now, divide both sides by 4:
\[ L = \frac{20}{4} = 5 \text{ cm} \]

### Step 5: Use the Pythagorean theorem to find the height (h)
In a right triangle formed by the radius, height, and slant height, we can use the Pythagorean theorem:
\[ L^2 = h^2 + r^2 \]
Substituting the known values:
\[ 5^2 = h^2 + 4^2 \]
This simplifies to:
\[ 25 = h^2 + 16 \]

### Step 6: Solve for h
Now, isolate h²:
\[ h^2 = 25 - 16 \]
\[ h^2 = 9 \]
Taking the square root:
\[ h = 3 \text{ cm} \]

### Step 7: Use the formula for the volume of a cone
The formula for the volume (V) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]

### Step 8: Substitute the values of r and h into the volume formula
Substituting the values we have:
\[ V = \frac{1}{3} \pi (4^2)(3) \]
\[ V = \frac{1}{3} \pi (16)(3) \]
\[ V = \frac{1}{3} \pi (48) \]
\[ V = 16\pi \text{ cm}^3 \]

### Final Answer
The volume of the conical vessel is \( 16\pi \text{ cm}^3 \).

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