Correct Option: A Step-by-step solution: 1. **Identify the given parameters:** * Base diameter of the cylindrical vessel $= 20\text{ cm}$ * Therefore, the radius of the cylindrical vessel, $r_c = \frac{20}{2} = 10\text{ cm}$ * Initial water height, $h_1 = 12\text{ cm}$ * Final water height after dropping balls, $h_2 = 13.6\text{ cm}$ * Number of identical spherical balls, $N = 15$ 2. **Calculate the rise in water level ($\\Delta h$):** The rise in water level is the difference between the final and initial heights. $\Delta h = h_2 - h_1 = 13.6\text{ cm} - 12\text{ cm} = 1.6\text{ cm}$ 3. **Apply the principle of water displacement:** When an object is submerged in a liquid, the volume of the liquid displaced is equal to the volume of the submerged object. In this case, the rise in water level is due to the total volume of the spherical balls. Volume of water displaced = Total volume of $N$ spherical balls 4. **Recall the relevant volume formulas:** * Volume of a cylinder = $\pi \times (\text{radius})^2 \times (\text{height})$ * Volume of a sphere = $\frac{4}{3} \times \pi \times (\text{radius})^3$ 5. **Set up the equation based on volume displacement:** Let $r_s$ be the radius of each spherical ball. The volume of water displaced is the volume of a cylindrical section with radius $r_c$ and height $\Delta h$. Volume of water displaced = $\pi r_c^2 \Delta h$ The total volume of $N$ spherical balls is: Total volume of $N$ spherical balls = $N \times \left(\frac{4}{3} \pi r_s^3\right)$ Equating the two volumes: $\pi r_c^2 \Delta h = N \times \frac{4}{3} \pi r_s^3$ 6. **Substitute the known values and solve for $r_s$:** $\pi (10)^2 (1.6) = 15 \times \frac{4}{3} \pi r_s^3$ Cancel $\pi$ from both sides of the equation: $(10)^2 (1.6) = 15 \times \frac{4}{3} r_s^3$ Simplify the terms: $100 \times 1.6 = (15 \times \frac{4}{3}) r_s^3$ $160 = (5 \times 4) r_s^3$ $160 = 20 r_s^3$ Divide both sides by $20$ to isolate $r_s^3$: $r_s^3 = \frac{160}{20}$ $r_s^3 = 8$ Take the cube root of both sides to find $r_s$: $r_s = \sqrt[3]{8}$ $r_s = 2\text{ cm}$ Thus, the radius of each spherical ball is $2\text{ cm}$. The final answer is $\boxed{2\text{ cm}}$.