Three solid spheres with radii r1, r2 and r3 are melted and recast into a single solid sphere. What is the radius of the new sphere expressed in terms of r1, r2 and r3?

Introduction / Context:
This is a symbolic version of the volume conservation idea used in mensuration. Instead of specific numerical radii, the radii r1, r2 and r3 are kept as variables and you must express the radius of the resulting sphere in terms of these variables. Such algebraic geometry questions appear in higher level aptitude and engineering entrance exams.

Given Data / Assumptions:
• Three solid spheres have radii r1, r2 and r3. • They are completely melted and recast into one solid sphere. • Volume is conserved during the melting and recasting process. • We must express the radius R of the new sphere in terms of r1, r2 and r3.

Concept / Approach:
The volume V of a sphere with radius r is V = (4 / 3) * π * r^3. When several spheres are melted together to form one new sphere, the total volume remains constant. Therefore, the sum of the volumes of the original three spheres will equal the volume of the new sphere of radius R. We write this equation using the general formula and then solve for R in terms of r1, r2 and r3. This will naturally lead to a cube root expression.

Step-by-Step Solution:
1. Volume of the first sphere: V1 = (4 / 3) * π * r1^3. 2. Volume of the second sphere: V2 = (4 / 3) * π * r2^3. 3. Volume of the third sphere: V3 = (4 / 3) * π * r3^3. 4. Total initial volume: V_total = V1 + V2 + V3 = (4 / 3) * π * (r1^3 + r2^3 + r3^3). 5. Let the radius of the new sphere be R. Then its volume is V_new = (4 / 3) * π * R^3. 6. By volume conservation, V_total = V_new. 7. So (4 / 3) * π * (r1^3 + r2^3 + r3^3) = (4 / 3) * π * R^3. 8. Cancel the common factor (4 / 3) * π from both sides to get r1^3 + r2^3 + r3^3 = R^3. 9. Take cube root on both sides: R = (r1^3 + r2^3 + r3^3)^(1/3).

Verification / Alternative check:
To verify the formula, you can test it with specific values. For example, if r1 = r2 = r3 = r, then the three equal spheres are melted into one sphere. The new volume is three times the original volume of one sphere, so R^3 = 3 * r^3 and R = (3^(1/3)) * r. Plugging r1 = r2 = r3 = r into the formula gives R = (r^3 + r^3 + r^3)^(1/3) = (3 * r^3)^(1/3) = (3^(1/3)) * r, which matches the direct reasoning.

Why Other Options Are Wrong:
• r1 + r2 + r3: This simply adds the radii linearly and ignores the cubic relationship of volume. • (r1^2 + r2^2 + r3^2)^(1/2): This is a root mean square style expression and fits area scaling, not volume scaling. • (r1^4 + r2^4 + r3^4)^(1/4): This relates to a fourth power and has no connection to the volume formula. • r1 * r2 * r3: Multiplying radii directly does not correspond to the sum of volumes and has no geometric meaning here.

Common Pitfalls:
A common mistake is to assume that the new radius is just the sum or average of the old radii. Another error is to forget that volume scales with the cube of the radius, not with r or r^2. Always start from the exact volume formula, sum the volumes correctly and then isolate R^3 before taking the cube root.

Final Answer:
The radius of the new sphere is (r1^3 + r2^3 + r3^3)^(1/3).