The circumference of the base of a 16 cm height solid cone is 33 cm. What is the volume of the cone in `cm^(3)` ?

Options

A. 1028
B. 616
C. 462
D. 828

Answer

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

### Step 1: Find the radius of the base of the cone.
The circumference \( C \) of the base of the cone is given by the formula:
\[ C = 2\pi r \]
We know that \( C = 33 \) cm. Therefore, we can set up the equation:
\[ 2\pi r = 33 \]
Using \( \pi \approx \frac{22}{7} \), we can substitute this value into the equation:
\[ 2 \times \frac{22}{7} \times r = 33 \]

### Step 2: Solve for \( r \).
First, simplify the equation:
\[ \frac{44}{7} r = 33 \]
Now, multiply both sides by \( \frac{7}{44} \) to isolate \( r \):
\[ r = 33 \times \frac{7}{44} \]
Calculating this gives:
\[ r = \frac{231}{44} = \frac{21}{4} \text{ cm} \]

### Step 3: Use the radius to find the volume of the cone.
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting \( r = \frac{21}{4} \) cm and \( h = 16 \) cm into the formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times \left(\frac{21}{4}\right)^2 \times 16 \]

### Step 4: Calculate \( r^2 \).
First, calculate \( \left(\frac{21}{4}\right)^2 \):
\[ \left(\frac{21}{4}\right)^2 = \frac{441}{16} \]

### Step 5: Substitute \( r^2 \) back into the volume formula.
Now substitute this back into the volume formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times \frac{441}{16} \times 16 \]
The \( 16 \) in the numerator and denominator cancels out:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 441 \]

### Step 6: Simplify the expression.
Now calculate:
\[ V = \frac{22 \times 441}{3 \times 7} \]
Calculating \( 3 \times 7 = 21 \):
\[ V = \frac{22 \times 441}{21} \]
Now divide \( 441 \) by \( 21 \):
\[ 441 \div 21 = 21 \]
So we have:
\[ V = 22 \times 21 = 462 \text{ cm}^3 \]

### Final Answer:
The volume of the cone is \( 462 \text{ cm}^3 \).

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