Each question consists of a question and two statements I and II given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and choose the appropriate option.
What is the respective ratio of volume of cylinder A and that of cylinder B?"
I The radius of cylinder A is 3/5 of radius of cylinder B and the height of cylinder A is 1/3 times the height of cylinder B.
II. The difference between volumes of cylinder A and, cylinder B is 7507.5 `m^3`.

Options

A. The data in statement I alone are sufficient to answer the question while the data in statement II alone are not sufficient to answer the question.
B. The data in statement II alone are sufficient to answer the question while the data in statement I alone are not sufficient to answer the question.
C. The data either in statement I alone or in statement II alone are sufficient to answer the question.
D. The datat in both the statements I and II together are necessary to answer the question.

Answer

To solve the problem of finding the respective ratio of the volumes of cylinder A and cylinder B based on the given statements, we will follow these steps:

### Step 1: Understand the Volume Formula
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Define Variables
Let:
- \( R_A \) = radius of cylinder A
- \( H_A \) = height of cylinder A
- \( R_B \) = radius of cylinder B
- \( H_B \) = height of cylinder B

### Step 3: Analyze Statement I
From Statement I:
- The radius of cylinder A is \( \frac{3}{5} \) of the radius of cylinder B:
\[ R_A = \frac{3}{5} R_B \]
- The height of cylinder A is \( \frac{1}{3} \) times the height of cylinder B:
\[ H_A = \frac{1}{3} H_B \]

### Step 4: Calculate the Volumes
Using the volume formula for both cylinders:
- Volume of cylinder A:
\[ V_A = \pi R_A^2 H_A = \pi \left(\frac{3}{5} R_B\right)^2 \left(\frac{1}{3} H_B\right) \]
Simplifying this:
\[ V_A = \pi \left(\frac{9}{25} R_B^2\right) \left(\frac{1}{3} H_B\right) = \frac{9\pi R_B^2 H_B}{75} \]

- Volume of cylinder B:
\[ V_B = \pi R_B^2 H_B \]

### Step 5: Find the Ratio of Volumes
Now, we find the ratio of the volumes \( \frac{V_A}{V_B} \):
\[ \frac{V_A}{V_B} = \frac{\frac{9\pi R_B^2 H_B}{75}}{\pi R_B^2 H_B} \]
Cancelling \( \pi R_B^2 H_B \) from the numerator and denominator:
\[ \frac{V_A}{V_B} = \frac{9}{75} = \frac{3}{25} \]

### Step 6: Conclusion from Statement I
Thus, from Statement I alone, we can conclude that the ratio of the volumes of cylinder A and cylinder B is \( 3:25 \).

### Step 7: Analyze Statement II
From Statement II:
- The difference between the volumes of cylinder A and cylinder B is given as \( 7507.5 \, m^3 \):
\[ V_B - V_A = 7507.5 \]
However, without knowing either volume specifically, we cannot derive the ratio from this statement alone.

### Final Conclusion
- Statement I alone is sufficient to answer the question.
- Statement II alone is not sufficient to answer the question.

### Answer
The correct option is that the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient.

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