From a right circular cylinder of radius 10 cm and height 21 cm, a right circular cone of same base radius is removed. If the volume of the remaining portion is `4400 cm^(3)`, then the height of the removed cone (taking `pi = (22)/(7)`) is :

Options

A. 15 cm
B. 18 cm
C. 21 cm
D. 24 cm

Answer

To find the height of the removed cone from the right circular cylinder, we can follow these steps:

### Step 1: Calculate the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Radius of the cylinder \( r = 10 \) cm
- Height of the cylinder \( H = 21 \) cm

Substituting these values into the formula:
\[ V_{\text{cylinder}} = \pi (10)^2 (21) = \pi \times 100 \times 21 = 2100\pi \, \text{cm}^3 \]

Using \( \pi = \frac{22}{7} \):
\[ V_{\text{cylinder}} = 2100 \times \frac{22}{7} = 6600 \, \text{cm}^3 \]

### Step 2: Set up the equation for the remaining volume
The volume of the remaining portion after removing the cone is given as \( 4400 \, \text{cm}^3 \). Therefore, we can write:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} \]
Substituting the known values:
\[ 4400 = 6600 - V_{\text{cone}} \]

### Step 3: Calculate the volume of the cone
Rearranging the equation gives:
\[ V_{\text{cone}} = 6600 - 4400 = 2200 \, \text{cm}^3 \]

The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

Using the same radius \( r = 10 \) cm for the cone, we can substitute:
\[ 2200 = \frac{1}{3} \pi (10)^2 h \]
\[ 2200 = \frac{1}{3} \pi \times 100 \times h \]
\[ 2200 = \frac{100\pi h}{3} \]

### Step 4: Substitute the value of \( \pi \) and solve for \( h \)
Substituting \( \pi = \frac{22}{7} \):
\[ 2200 = \frac{100 \times \frac{22}{7} \times h}{3} \]
Multiplying both sides by 3:
\[ 6600 = \frac{2200h}{7} \]
Multiplying both sides by 7:
\[ 46200 = 2200h \]
Dividing both sides by 2200:
\[ h = \frac{46200}{2200} = 21 \, \text{cm} \]

### Final Answer
The height of the removed cone is \( 21 \, \text{cm} \).

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