The height of a cone is 7 cm and diameter of its base is 14 cm. What is the volume of the cone?

Options

A. `411. 6 cm ^(3) `
B. `359. 3 cm ^(3)`
C. `442.6 cm ^(3)`
D. `450.6 cm ^(3)`

Answer

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

### Step 1: Identify the given values
- Height (h) of the cone = 7 cm
- Diameter of the base = 14 cm

### Step 2: Calculate the radius of the base
The radius (r) is half of the diameter.
\[ r = \frac{\text{Diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm} \]

### Step 3: 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 4: Substitute the values into the formula
Now, we will substitute the values of radius and height into the formula:
\[ V = \frac{1}{3} \pi (7 \text{ cm})^2 (7 \text{ cm}) \]

### Step 5: Calculate \( r^2 \)
\[ r^2 = (7 \text{ cm})^2 = 49 \text{ cm}^2 \]

### Step 6: Substitute \( r^2 \) back into the volume formula
\[ V = \frac{1}{3} \pi (49 \text{ cm}^2) (7 \text{ cm}) \]

### Step 7: Multiply the values
\[ V = \frac{1}{3} \pi (343 \text{ cm}^3) \]

### Step 8: Use the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times 343 \text{ cm}^3 \]

### Step 9: Simplify the expression
Calculating \( \frac{22 \times 343}{3 \times 7} \):
\[ V = \frac{22 \times 343}{21} \]

### Step 10: Calculate \( 22 \times 343 \)
\[ 22 \times 343 = 7566 \]
So,
\[ V = \frac{7566}{21} \text{ cm}^3 \]

### Step 11: Divide to find the volume
Calculating \( \frac{7566}{21} \):
\[ V = 360.2857 \text{ cm}^3 \approx 360.3 \text{ cm}^3 \]

### Final Answer:
The volume of the cone is approximately \( 360.3 \text{ cm}^3 \).

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