Find the volume (in `cm^(3)`) of a right circular cone of diameter 7 cm and height 7 cm.

Options

A. 69.06
B. 16.43
C. 11.15
D. 89.83

Answer

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

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

where:
- \( V \) is the volume,
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone,
- \( \pi \) is a constant approximately equal to \( \frac{22}{7} \) or \( 3.14 \).

### Step-by-Step Solution:

**Step 1: Find the radius of the cone.**
- Given the diameter of the cone is 7 cm.
- The radius \( r \) is half of the diameter.

\[ r = \frac{\text{diameter}}{2} = \frac{7 \text{ cm}}{2} = 3.5 \text{ cm} \]

**Step 2: Identify the height of the cone.**
- The height \( h \) of the cone is given as 7 cm.

**Step 3: Substitute the values into the volume formula.**
- Now we can substitute \( r \) and \( h \) into the volume formula.

\[ V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (3.5)^2 (7) \]

**Step 4: Calculate \( r^2 \).**
- First, calculate \( r^2 \):

\[ r^2 = (3.5)^2 = 12.25 \]

**Step 5: Substitute \( r^2 \) back into the volume formula.**
- Now substitute \( r^2 \) into the volume formula:

\[ V = \frac{1}{3} \pi (12.25)(7) \]

**Step 6: Calculate \( 12.25 \times 7 \).**
- Now calculate the product:

\[ 12.25 \times 7 = 85.75 \]

**Step 7: Substitute \( 85.75 \) into the volume formula.**
- Substitute this value back into the volume formula:

\[ V = \frac{1}{3} \pi (85.75) \]

**Step 8: Use \( \pi \approx \frac{22}{7} \) to calculate the volume.**
- Now, substitute \( \pi \):

\[ V = \frac{1}{3} \times \frac{22}{7} \times 85.75 \]

**Step 9: Calculate the final volume.**
- First, calculate \( \frac{22 \times 85.75}{7} \):

\[ \frac{22 \times 85.75}{7} = \frac{1886.5}{7} \approx 269.5 \]

- Now, divide by 3:

\[ V \approx \frac{269.5}{3} \approx 89.83 \text{ cm}^3 \]

### Final Answer:
The volume of the right circular cone is approximately \( 89.83 \text{ cm}^3 \).