Assertion (A) : If surface areas of the two spheres are in the ratio 16 : 9, then their volumes are in the ratio 64 : 27. Reason (R) : If $S_1$ and $S_2$ are the surface areas of two spheres and $V_1$ and $V_2$ are their volumes respectively, then $\frac{V_1}{V_2} = \left(\frac{S_1}{S_2}\right)^{3/2}$.

Step 1: Relate surface area and radius

The surface area of a sphere is given by the formula $S = 4\pi r^2$, where $r$ is the radius of the sphere. If we have two spheres with radii $r_1$ and $r_2$, their surface areas $S_1$ and $S_2$ will be $S_1 = 4\pi r_1^2$ and $S_2 = 4\pi r_2^2$.

Step 2: Find the ratio of radii

Given the ratio of surface areas $\frac{S_1}{S_2} = \frac{16}{9}$, we can find the ratio of their radii. By dividing the surface area formulas, we get $\frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2$. Taking the square root of both sides gives us the ratio of the radii.

Step 3: Relate volume and radius

The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$. For two spheres with radii $r_1$ and $r_2$, their volumes $V_1$ and $V_2$ will be $V_1 = \frac{4}{3}\pi r_1^3$ and $V_2 = \frac{4}{3}\pi r_2^3$.

Step 4: Find the ratio of volumes

We can find the ratio of the volumes by dividing their formulas. This simplifies to $\frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3$.

Step 5: Substitute ratio of radii into ratio of volumes

Now we substitute the expression for the ratio of radii, $\frac{r_1}{r_2} = \sqrt{\frac{S_1}{S_2}}$, into the ratio of volumes. This gives us $\frac{V_1}{V_2} = \left(\frac{S_1}{S_2}\right)^{3/2}$. This confirms that Reason (R) is a correct statement.

Step 6: Evaluate the assertion

Given that the ratio of surface areas is $16:9$, we can use the derived relationship to find the ratio of volumes. Substituting $\frac{S_1}{S_2} = \frac{16}{9}$ into the formula, we get $\frac{V_1}{V_2} = \left(\frac{16}{9}\right)^{3/2}$. This evaluates to $\frac{64}{27}$, which means the Assertion (A) is also correct.

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