The curved surface area of a right circular cone is 65 `pi cm^(2)` and the radius of its base is 5 cm. What is 40% of the volume of the cone, in `cm^(3)` ?

Options

A. `50pi`
B. `40pi`
C. `180pi`
D. `100pi`

Answer

To find 40% of the volume of a right circular cone given its curved surface area and radius, we can follow these steps:

### Step 1: Understand the given data
- Curved Surface Area (CSA) of the cone = \( 65 \pi \, \text{cm}^2 \)
- Radius (r) of the base = \( 5 \, \text{cm} \)

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

### Step 3: Substitute the known values into the formula
Substituting the given values into the formula:
\[ 65 \pi = \pi \cdot 5 \cdot l \]

### Step 4: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ 65 = 5l \]

### Step 5: Solve for the slant height (l)
Now, divide both sides by 5:
\[ l = \frac{65}{5} = 13 \, \text{cm} \]

### Step 6: Use the Pythagorean theorem to find the height (h)
Using the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + h^2 \]
\[ 169 = 25 + h^2 \]
\[ h^2 = 169 - 25 = 144 \]
Taking the square root:
\[ h = \sqrt{144} = 12 \, \text{cm} \]

### Step 7: Calculate the volume of the cone
The formula for the volume (V) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the known values:
\[ V = \frac{1}{3} \pi (5^2)(12) \]
\[ V = \frac{1}{3} \pi (25)(12) \]
\[ V = \frac{1}{3} \pi (300) \]
\[ V = 100 \pi \, \text{cm}^3 \]

### Step 8: Calculate 40% of the volume
To find 40% of the volume:
\[ 40\% \text{ of } V = 0.4 \times 100 \pi = 40 \pi \, \text{cm}^3 \]

### Final Answer:
Thus, 40% of the volume of the cone is:
\[ \boxed{40 \pi \, \text{cm}^3} \]