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

Options

A. `50pi`
B. `100pi`
C. `180pi`
D. `45pi`

Answer

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

### Step 1: Understand the formulas
The curved surface area (CSA) of a cone is given by the formula:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius of the base and \( l \) is the slant height of the cone.

The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( h \) is the height of the cone.

### Step 2: Substitute the given values into the CSA formula
We know the curved surface area is \( 65\pi \) cm² and the radius \( r = 5 \) cm. Substituting these values into the CSA formula:
\[ 65\pi = \pi \cdot 5 \cdot l \]
We can simplify this equation by dividing both sides by \( \pi \):
\[ 65 = 5l \]

### Step 3: Solve for the slant height \( l \)
Now, we can solve for \( l \):
\[ l = \frac{65}{5} = 13 \text{ cm} \]

### Step 4: Use the Pythagorean theorem to find the height \( h \)
In a right circular cone, the relationship between the radius \( r \), height \( h \), and slant height \( l \) is given by:
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 13^2 = 5^2 + h^2 \]
This simplifies to:
\[ 169 = 25 + h^2 \]
\[ h^2 = 169 - 25 = 144 \]
Taking the square root:
\[ h = \sqrt{144} = 12 \text{ cm} \]

### Step 5: Calculate the volume \( V \) of the cone
Now we can calculate the volume of the cone using the volume formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting \( r = 5 \) cm and \( h = 12 \) cm:
\[ V = \frac{1}{3} \pi (5^2)(12) = \frac{1}{3} \pi (25)(12) = \frac{300}{3} \pi = 100\pi \text{ cm}^3 \]

### Step 6: Find half of the volume
To find half of the volume:
\[ \text{Half of the volume} = \frac{1}{2} V = \frac{1}{2} (100\pi) = 50\pi \text{ cm}^3 \]

### Final Answer
Thus, half of the volume of the cone is:
\[ \boxed{50\pi \text{ cm}^3} \]