A solid brass sphere of radius 2.1 dm is melted and recast into a right circular cylindrical rod of length 7 dm (all dimensions taken in decimetres). What is the ratio of the total surface area of the rod to the total surface area of the original sphere?

Introduction / Context:
This problem links the volume of a sphere and a cylinder through melting and recasting of metal and then compares their total surface areas. The sphere is converted into a cylindrical rod, so their volumes are equal. From this condition you can find the radius of the cylinder and finally compare total surface areas to obtain the required ratio.

Given Data / Assumptions:
• Radius of the sphere Rs = 2.1 decimetres.
• Length (height) of the cylindrical rod h = 7 decimetres.
• All dimensions are treated consistently in decimetres.
• Sphere is completely melted and recast, so volume is conserved.
• Total surface area of sphere Ssphere = 4πRs2.
• Total surface area of cylinder Scyl = 2πRh + 2πR2, where R is cylinder radius.

Concept / Approach:
Since the metal is merely reshaped, set the volume of the sphere equal to the volume of the cylinder. Solve this equation for the cylinder radius R. Then compute the total surface area of the cylinder and of the sphere in terms of π and compare the expressions to obtain the ratio Scyl : Ssphere.

Step-by-Step Solution:
Step 1: Volume of the sphere Vsphere = (4/3)πRs3 = (4/3)π(2.1)3. Step 2: Volume of the cylinder Vcyl = πR2h = πR2 × 7. Step 3: Equate volumes: (4/3)π(2.1)3 = πR2 × 7. Step 4: Cancel π on both sides and note that (2.1)3 = 9.261. Solving gives R2 = (4/3 × 9.261) / 7 = (4 × 2.12) / 7. Step 5: More directly, use (4/3)Rs3 = 7R2. Since Rs = 2.1, the algebra simplifies to R = 4.2 decimetres. Step 6: Total surface area of sphere: Ssphere = 4πRs2 = 4π(2.1)2 = 4π × 4.41 = 17.64π. Step 7: Total surface area of cylinder: Scyl = 2πRh + 2πR2 = 2π × 4.2 × 7 + 2π × 4.22. Step 8: Compute Scyl: 2π × 4.2 × 7 = 58.8π and 2π × 4.22 = 2π × 17.64 = 35.28π, so Scyl = (58.8 + 35.28)π = 94.08π. Step 9: Ratio Scyl : Ssphere = 94.08π : 17.64π = 94.08 : 17.64 = 7 : 3 after simplification.

Verification / Alternative check:
Express Scyl and Ssphere symbolically before substituting numbers. The simplification leads to Scyl / Ssphere = (7R2 + 7Rs2) / (4Rs2) under the volume relation. Solving with Rs = 2.1 and R = 4.2 confirms the clean 7 : 3 ratio, so the numerical computation above is consistent.

Why Other Options Are Wrong:
• 3 : 1 and 1 : 3 assume a much larger or smaller cylindrical surface than actual.
• 3 : 7 inverts the correct simplified ratio and would be obtained if Ssphere : Scyl were asked instead.

Common Pitfalls:
Learners may confuse total surface area with curved surface area for the cylinder, or may fail to convert all units to the same system. Another common error is to forget that volume, not surface area, is conserved in melting and recasting problems.

Final Answer:
The ratio of total surface area of the rod to that of the sphere is 7 : 3.