A lead pencil it is the shape of a cylinder. The pencil is 21 cm long with radius `0.4` cm and its lead is of radius `0.1` cm. What is the volume of wood in the pencil ?

Answer

To find the volume of wood in the pencil, we need to calculate the volume of the entire pencil and then subtract the volume of the lead inside it.

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

For the pencil:
- Radius \( r = 0.4 \) cm
- Height \( h = 21 \) cm

Substituting the values into the formula:
\[ V_{\text{pencil}} = \pi (0.4)^2 (21) \]
\[ = \pi (0.16) (21) \]
\[ = \pi (3.36) \]
\[ \approx 10.55 \text{ cm}^3 \quad (\text{using } \pi \approx 3.14) \]

### Step 2: Calculate the volume of the lead (inner cylinder)
Now, we calculate the volume of the lead, which is also in the shape of a cylinder.

For the lead:
- Radius \( r = 0.1 \) cm
- Height \( h = 21 \) cm

Using the same formula:
\[ V_{\text{lead}} = \pi (0.1)^2 (21) \]
\[ = \pi (0.01) (21) \]
\[ = \pi (0.21) \]
\[ \approx 0.66 \text{ cm}^3 \quad (\text{using } \pi \approx 3.14) \]

### Step 3: Calculate the volume of the wood
To find the volume of the wood in the pencil, we subtract the volume of the lead from the volume of the pencil:
\[ V_{\text{wood}} = V_{\text{pencil}} - V_{\text{lead}} \]
\[ = 10.55 \text{ cm}^3 - 0.66 \text{ cm}^3 \]
\[ = 9.89 \text{ cm}^3 \]

### Final Answer
The volume of wood in the pencil is approximately \( 9.89 \text{ cm}^3 \).

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