Assertion (R) : If the volume of two spheres are in the ratio 27 : 8. Then their surface area are in the ratio 3:2.
Reason (R) : Volume of sphere `= (4)/(3)pi r^(3)` and its surface area `= 4pi r^(2)`.

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is not the correct explanation of A.
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem, we need to analyze the assertion and reason provided in the question step by step.

### Step 1: Understanding the Volume Ratio
Given that the volumes of two spheres are in the ratio \(27 : 8\), we can express this mathematically. The volume \(V\) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Let the radii of the two spheres be \(R\) and \(r\). Therefore, we have:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi R^3}{\frac{4}{3} \pi r^3} = \frac{R^3}{r^3} \]
According to the problem, this ratio is equal to \(27 : 8\):
\[ \frac{R^3}{r^3} = \frac{27}{8} \]

### Step 2: Finding the Ratio of Radii
To find the ratio of the radii, we take the cube root of both sides:
\[ \frac{R}{r} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} \]

### Step 3: Calculating the Surface Area Ratio
Next, we need to find the ratio of the surface areas of the two spheres. The surface area \(A\) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]
Thus, the ratio of the surface areas of the two spheres is:
\[ \frac{A_1}{A_2} = \frac{4 \pi R^2}{4 \pi r^2} = \frac{R^2}{r^2} \]
Substituting the ratio of the radii we found earlier:
\[ \frac{R^2}{r^2} = \left(\frac{R}{r}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Step 4: Conclusion
The assertion states that the surface areas are in the ratio \(3 : 2\), which is equivalent to \(\frac{3}{2}\). However, we found that the ratio of the surface areas is actually \(\frac{9}{4}\). Therefore, the assertion is false.

### Final Answer
- Assertion (A) is false.
- Reason (R) is true.