A spherical ball of lead, $3$ cm in diameter is melted and recast into three spherical balls. The diameter of two of these are $1.5$ cm and $2$ cm respectively. The diameter of the third ball is

### Concept & Conservation of Volume (Using Diameters) The volume of a sphere is proportional to the cube of its diameter ($V = \frac{\pi}{6}d^3$). When a sphere is melted into smaller ones, the sum of the cubes of the diameters of the smaller spheres equals the cube of the diameter of the original sphere. $$D^3 = d_1^3 + d_2^3 + d_3^3$$ ### Step-by-Step Solution * **Given:** Original diameter $D = 3$ cm. Diameters of two smaller balls: $d_1 = 1.5$ cm, $d_2 = 2$ cm. * Let the diameter of the third ball be $d_3$. * $3^3 = 1.5^3 + 2^3 + d_3^3$ * $27 = 3.375 + 8 + d_3^3$ * $27 = 11.375 + d_3^3$ * $d_3^3 = 27 - 11.375 = 15.625$ * To find the cube root of $15.625$, convert it to a fraction: $15.625 = \frac{15625}{1000}$. * $d_3 = \sqrt[3]{\frac{15625}{1000}} = \frac{25}{10} = 2.5$ cm. ### Exam Strategy & Shortcut By halving the standard $3-4-5 \rightarrow 6$ cubic relation (from $3^3+4^3+5^3 = 6^3$), we see that $(1.5)^3 + (2)^3 + (2.5)^3 = 3^3$. Recognizing scalar multiples of famous volume triplets saves you from decimal arithmetic completely. ### Common Pitfall Converting diameters to radii before cubing. While mathematically correct, it introduces unnecessary fractions and complex arithmetic. Cubing diameters directly works because the proportionality constant ($\frac{\pi}{6}$) cancels out on both sides. ### Final Answer Therefore, the correct answer is **2.5 cm**.