Answer

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**Hint:**

Here we need to find the number of refills that can be filled with the given amount of ink. First, we will find the volume of ink filled in the cylindrical bottle and then we will find the volume of ink filled in each ball pen refill. We will then find the number of refills that can be filled, which will be equal to the ratio of volume of ink filled in the cylindrical bottle to the volume of ink filled in each ball pen refill. From there, we will get our final result.

**Formula used:**

Volume of cylinder$ = \pi {r^2}h$, where $r$ is the radius of the cylinder and $h$ is the height of the cylinder.

**Complete step by step solution:**

We will first find the volume of the cylindrical container.

Using the formula of volume of cylinder, we can say that

Volume of cylindrical container $ = \pi {r^2}h$

Now, substituting $r = 6$ and $h = 14$ in the above equation, we get

$ \Rightarrow $ Volume of cylindrical container $ = \pi \cdot {6^2} \cdot 14$

But, it is given that the volume of ink filled in the cylindrical container $71\% $ of the volume of the cylindrical container.

Therefore, we get

Volume of ink filled in the cylindrical container $ = \dfrac{{71}}{{100}} \times \pi \times 6 \times 6 \times 14$ …………. $\left( 1 \right)$

Now, it is also given that the length of the ball pen refill is equal to 12 cm and the diameter of the ball pen refill is equal to 2 mm.

Therefore, the radius of the pen refill in centimeter $ = \dfrac{2}{2} \times \dfrac{1}{{10}} = 0.1cm$

Now, we will find the volume of the ball pen refill.

Volume of ball pen refill $ = \pi {R^2}H$

Now, substituting $R = 0.1$ and $H = 12$ in the above equation, we get

$ \Rightarrow $ Volume of ball pen refill $ = \pi \cdot {\left( {0.1} \right)^2} \cdot 12$

But, the volume of ink filled in the ball pen refill $84\% $ of the volume of the ball pen refill.

Therefore, we get

Volume of ink filled in the ball pen refill $ = \dfrac{{84}}{{100}} \times \pi \times 0.1 \times 0.1 \times 12$ …………. $\left( 2 \right)$

Now, we will find the number of refills that can be filled with this ink.

The number of refilled that can be filled will be equal to the ratio of volume of ink filled in the cylindrical bottle to the volume of ink filled in each ball pen refill. Therefore,

Number of pen refill $ = \dfrac{{\dfrac{{71}}{{100}} \times \pi \times 6 \times 6 \times 14}}{{\dfrac{{84}}{{100}} \times \pi \times 0.1 \times 0.1 \times 12}}$

On further simplification, we get

$ \Rightarrow $ Number of pen refill \[ = \dfrac{{71}}{{100}} \times 6 \times 6 \times 14 \times \dfrac{{100}}{{84}} \times \dfrac{{100}}{{12}}\]

On multiplying the terms, we get

$ \Rightarrow $ Number of pen refill \[ = 3550\]

**Hence, the required number of pen refills that can be filled is equal to 3550.**

**Note:**

Volume of any object is defined as the amount of space that an object occupies. But to calculate the amount of space available in a hollow cylinder, we always consider the inner radius of the hollow cylinder.

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