A cylinder with height and radius 2:1 is filled with soft drink and then it is titled so as to allow some soft drink to flow off to an extent where the level of soft drink just touches the lowest point of the upper mouth .
If the quantity of soft drink left is poured into a conical flask whose heights and base radius are same as that of the cylinder so as to fill the conical flask completely , the quantity of soft drink left in the cylinder as a fraction of its total capacity is :

Options

A. `1/3`
B. `1/6`
C. `1/9`
D. `1/10`

Answer

To solve the problem, we need to determine the fraction of soft drink left in the cylinder after some has been poured into a conical flask. Let's break it down step by step.

### Step 1: Understand the dimensions of the cylinder
Given that the height (H) and radius (R) of the cylinder are in the ratio 2:1, we can express them as:
- Height of the cylinder, H = 2x
- Radius of the cylinder, R = x

### Step 2: Calculate the volume of the cylinder
The formula for the volume of a cylinder is given by:
\[ V_{cylinder} = \pi R^2 H \]
Substituting the values of R and H:
\[ V_{cylinder} = \pi (x^2)(2x) = 2\pi x^3 \]

### Step 3: Understand the effect of tilting the cylinder
When the cylinder is tilted, the soft drink level touches the lowest point of the upper mouth. This means that half of the volume of the cylinder is effectively lost. Therefore, the remaining volume of soft drink in the cylinder is:
\[ V_{remaining} = \frac{1}{2} V_{cylinder} = \frac{1}{2} (2\pi x^3) = \pi x^3 \]

### Step 4: Calculate the volume of the conical flask
The conical flask has the same height and base radius as the cylinder. The formula for the volume of a cone is:
\[ V_{cone} = \frac{1}{3} \pi R^2 H \]
Substituting the values of R and H:
\[ V_{cone} = \frac{1}{3} \pi (x^2)(2x) = \frac{2}{3} \pi x^3 \]

### Step 5: Determine how much soft drink can fill the conical flask
The remaining soft drink in the cylinder, which is \( \pi x^3 \), is poured into the conical flask. Since the volume of the cone is \( \frac{2}{3} \pi x^3 \), we can see that:
- The conical flask can hold \( \frac{2}{3} \pi x^3 \) of soft drink.
- The remaining soft drink after filling the cone will be:
\[ V_{remaining\_after\_pouring} = V_{remaining} - V_{cone} = \pi x^3 - \frac{2}{3} \pi x^3 = \frac{1}{3} \pi x^3 \]

### Step 6: Calculate the fraction of soft drink left in the cylinder
Now, we need to find the fraction of the remaining soft drink in the cylinder with respect to its total capacity:
\[ \text{Fraction} = \frac{V_{remaining\_after\_pouring}}{V_{cylinder}} = \frac{\frac{1}{3} \pi x^3}{2 \pi x^3} = \frac{1}{6} \]

### Final Answer
The quantity of soft drink left in the cylinder as a fraction of its total capacity is \( \frac{1}{6} \).

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