The radius of the base of a right circular cone is 6 cm and its slant height is 10 cm. Then its volume is (Use `pi = (22)/(7)`)

Options

A. a) `301.71 cm^(3)`
B. b) `310.71 cm^(3)`
C. c) `301.17 cm^(3)`
D. d) `310.17 cm^(3)`

Answer

To find the volume of a right circular cone, we can use the formula:

\[ \text{Volume} = \frac{1}{3} \pi r^2 h \]

where:
- \( r \) is the radius of the base,
- \( h \) is the height of the cone,
- \( \pi \) is approximately \( \frac{22}{7} \).

### Step 1: Identify the given values
- Radius \( r = 6 \, \text{cm} \)
- Slant height \( l = 10 \, \text{cm} \)

### Step 2: Find the height of the cone
To find the height \( h \) of the cone, we can use the Pythagorean theorem. The relationship between the radius, height, and slant height is given by:

\[ l^2 = r^2 + h^2 \]

Substituting the known values:

\[ 10^2 = 6^2 + h^2 \]

Calculating the squares:

\[ 100 = 36 + h^2 \]

Rearranging to find \( h^2 \):

\[ h^2 = 100 - 36 = 64 \]

Taking the square root to find \( h \):

\[ h = \sqrt{64} = 8 \, \text{cm} \]

### Step 3: Substitute the values into the volume formula
Now that we have \( h \), we can substitute \( r \) and \( h \) into the volume formula:

\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times (6^2) \times 8 \]

Calculating \( 6^2 \):

\[ 6^2 = 36 \]

Now substituting back into the volume formula:

\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times 36 \times 8 \]

### Step 4: Simplify the expression
Calculating the multiplication:

\[ 36 \times 8 = 288 \]

Now substituting this back into the volume formula:

\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times 288 \]

Calculating \( \frac{288}{3} \):

\[ \frac{288}{3} = 96 \]

Now substituting this value:

\[ \text{Volume} = \frac{22}{7} \times 96 \]

### Step 5: Final calculation
Calculating \( 22 \times 96 \):

\[ 22 \times 96 = 2112 \]

Now dividing by 7:

\[ \text{Volume} = \frac{2112}{7} \approx 301.71 \, \text{cm}^3 \]

### Conclusion
Thus, the volume of the cone is approximately:

\[ \text{Volume} \approx 301.71 \, \text{cm}^3 \]