The circumference of the base of a right circular cylinder is 88 cm. If the height of the cylinder is 21 cm, what is the volume of a right circular cone whose height and radius are equal to that of the cylinder ? (in cm3)

Options

A. 4312
B. 3402
C. 3436
D. 4364

Answer

To find the volume of a right circular cone whose height and radius are equal to that of a given right circular cylinder, we can follow these steps:

### Step 1: Find the radius of the cylinder
The circumference (C) of the base of the cylinder is given as 88 cm. The formula for the circumference of a circle is:
\[ C = 2\pi r \]
Where \( r \) is the radius.

We can rearrange this formula to solve for \( r \):
\[ r = \frac{C}{2\pi} \]

### Step 2: Substitute the values
Substituting the given circumference into the formula:
\[ r = \frac{88}{2\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{88}{2 \times \frac{22}{7}} \]
\[ r = \frac{88 \times 7}{44} \]
\[ r = \frac{616}{44} \]
\[ r = 14 \text{ cm} \]

### Step 3: Identify the height of the cone
The height (h) of the cylinder is given as 21 cm. Since the height of the cone is equal to that of the cylinder, we have:
\[ h = 21 \text{ cm} \]

### Step 4: Calculate the volume of the cone
The formula for the volume (V) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]

### Step 5: Substitute the values into the volume formula
Substituting the values of \( r \) and \( h \):
\[ V = \frac{1}{3} \pi (14)^2 (21) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times 196 \times 21 \]

### Step 6: Simplify the expression
Calculating \( 14^2 \):
\[ 14^2 = 196 \]
Now substituting back:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 196 \times 21 \]

### Step 7: Calculate the volume
First, we can simplify \( \frac{196 \times 21}{3} \):
\[ \frac{196 \times 21}{3} = \frac{4116}{3} = 1372 \]

Now substituting back:
\[ V = \frac{22}{7} \times 1372 \]

### Step 8: Final calculation
Calculating \( \frac{22 \times 1372}{7} \):
\[ 22 \times 1372 = 30284 \]
Now dividing by 7:
\[ V = \frac{30284}{7} = 4312 \text{ cm}^3 \]

### Final Answer
The volume of the right circular cone is \( 4312 \text{ cm}^3 \).

---