The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used up on writing 330 words on an average. How many words would use up a bottle of ink containing one fifth of a litre ?

Options

A. 60000
B. 66000
C. 48000
D. 50000

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert the diameter to radius
Given the diameter of the barrel is 5 mm, we can find the radius by dividing the diameter by 2.

\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{5 \text{ mm}}{2} = 2.5 \text{ mm} \]

### Step 2: Convert the radius to centimeters
Since the height of the barrel is given in centimeters, we need to convert the radius from millimeters to centimeters.

\[ \text{Radius in cm} = \frac{2.5 \text{ mm}}{10} = 0.25 \text{ cm} \]

### Step 3: Calculate the volume of the barrel
The formula for the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius,
- \( h \) is the height,
- \( \pi \) can be approximated as \( \frac{22}{7} \).

Substituting the values:

\[ V = \frac{22}{7} \times (0.25)^2 \times 7 \]

Calculating \( (0.25)^2 \):

\[ (0.25)^2 = 0.0625 \]

Now substituting back into the volume formula:

\[ V = \frac{22}{7} \times 0.0625 \times 7 \]

The \( 7 \) in the numerator and denominator cancels out:

\[ V = 22 \times 0.0625 = 1.375 \text{ cm}^3 \]

### Step 4: Determine words per cm³ of ink
We know that a full barrel of ink allows writing 330 words. Therefore, the number of words per cm³ of ink is:

\[ \text{Words per cm}^3 = \frac{330 \text{ words}}{1.375 \text{ cm}^3} \approx 240 \text{ words/cm}^3 \]

### Step 5: Convert one-fifth of a litre to cm³
1 litre is equal to 1000 cm³. Therefore, one-fifth of a litre is:

\[ \frac{1000 \text{ cm}^3}{5} = 200 \text{ cm}^3 \]

### Step 6: Calculate total words for 200 cm³ of ink
Now, we can calculate how many words can be written with 200 cm³ of ink:

\[ \text{Total words} = 240 \text{ words/cm}^3 \times 200 \text{ cm}^3 = 48000 \text{ words} \]

### Final Answer
Thus, a bottle of ink containing one-fifth of a litre can be used to write **48000 words**.

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