From a solid cylinder of height 14cm and base diameter 7cm,two equal conical holes each of radius 2.1cm and height 4cm are cut off. Find the volume of remmaining solid.

Answer

To find the volume of the remaining solid after cutting two equal conical holes from a solid cylinder, we will follow these steps:

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

Given:
- Height of the cylinder \( h = 14 \, \text{cm} \)
- Diameter of the base \( = 7 \, \text{cm} \) so the radius \( r = \frac{7}{2} = 3.5 \, \text{cm} \)

Substituting the values into the formula:
\[ V = \pi (3.5)^2 (14) \]
\[ = \pi (12.25)(14) \]
\[ = \pi (171.5) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V \approx \frac{22}{7} \times 171.5 = 22 \times 24.5 = 539 \, \text{cm}^3 \]

### Step 2: Calculate the volume of one conical hole
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.

Given:
- Radius of the cone \( r = 2.1 \, \text{cm} \)
- Height of the cone \( h = 4 \, \text{cm} \)

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (2.1)^2 (4) \]
\[ = \frac{1}{3} \pi (4.41)(4) \]
\[ = \frac{1}{3} \pi (17.64) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V \approx \frac{1}{3} \times \frac{22}{7} \times 17.64 \]
Calculating:
\[ = \frac{22 \times 17.64}{21} \approx \frac{388.08}{21} \approx 18.46 \, \text{cm}^3 \]

### Step 3: Calculate the total volume of the two conical holes
Since there are two conical holes, we multiply the volume of one cone by 2:
\[ \text{Total volume of conical holes} = 2 \times 18.46 \approx 36.92 \, \text{cm}^3 \]

### Step 4: Calculate the volume of the remaining solid
Now, we subtract the total volume of the conical holes from the volume of the cylinder:
\[ \text{Volume of remaining solid} = \text{Volume of cylinder} - \text{Total volume of conical holes} \]
\[ = 539 - 36.92 \approx 502.08 \, \text{cm}^3 \]

### Final Answer
The volume of the remaining solid is approximately \( 502.08 \, \text{cm}^3 \).

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