A solid hemisphere of radius 14 cm is melted and recast to form a closed cylinder. For this cylinder, the ratio of its curved surface area (CSA) to its total surface area (TSA) is 2 : 3. What is the radius, in centimetres (correct to one decimal place), of the base of the cylinder?

Introduction / Context:
This question combines volume conservation with surface area relationships. A hemisphere is melted and recast into a closed cylinder. Since the material is only reshaped, the volume remains constant, but the dimensions of the new solid must satisfy an additional condition relating its curved surface area (CSA) and total surface area (TSA). The problem illustrates how different solid geometry formulas interact in advanced aptitude questions.

Given Data / Assumptions:

• Original solid: a hemisphere of radius 14 cm.
• New solid: a closed cylinder (curved surface plus two circular ends).
• Volume of hemisphere = volume of cylinder.
• For the cylinder, CSA : TSA = 2 : 3.
• We must find the radius R of the cylinder base, correct to one decimal place.
• Use π consistently (the value cancels in most ratios).

Concept / Approach:
We use three formulas:

• Volume of hemisphere: V = (2/3) * π * r^3, where r = 14 cm.
• Volume of cylinder: V = π * R^2 * h, where R is its base radius and h is its height.
• For a closed cylinder, curved surface area CSA = 2πRh and total surface area TSA = CSA + 2πR^2.

The given ratio CSA : TSA = 2 : 3 provides one relation between h and R, and volume equality provides another. Solving these equations together gives the radius R.

Step-by-Step Solution:
Step 1: Express the CSA and TSA of the cylinder. CSA = 2πRh TSA = 2πRh + 2πR^2 Step 2: Use the given ratio CSA : TSA = 2 : 3. (2πRh) : (2πRh + 2πR^2) = 2 : 3 Cancel common factors 2πR: Rh : (Rh + R^2) = 2 : 3 3Rh = 2Rh + 2R^2 ⇒ Rh = 2R^2 ⇒ h = 2R Step 3: Use volume equality. Volume of hemisphere = volume of cylinder. Hemisphere: V = (2/3) * π * 14^3 Cylinder: V = π * R^2 * h = π * R^2 * (2R) = 2πR^3 Step 4: Set volumes equal and simplify. 2πR^3 = (2/3) * π * 14^3 Cancel 2π: R^3 = (1/3) * 14^3 R = 14 / (3^(1/3)) Step 5: Approximate the cube root of 3. 3^(1/3) ≈ 1.442 Therefore R ≈ 14 / 1.442 ≈ 9.7 cm (to one decimal place).

Verification / Alternative check:
We can approximate h = 2R. With R ≈ 9.7 cm, h ≈ 19.4 cm. Compute CSA ≈ 2πRh ≈ 2 * π * 9.7 * 19.4 and TSA ≈ CSA + 2πR^2. Using π ≈ 3.14 gives CSA ≈ 2 * 3.14 * 9.7 * 19.4 ≈ 1181.9 and 2πR^2 ≈ 2 * 3.14 * 9.7^2 ≈ 590.6. Thus TSA ≈ 1181.9 + 590.6 ≈ 1772.5. The ratio CSA / TSA ≈ 1181.9 / 1772.5 ≈ 0.667, which is very close to 2/3, confirming that R ≈ 9.7 cm is consistent with the condition.

Why Other Options Are Wrong:
Option 2: 10.3 cm would produce a different ratio of CSA to TSA and a cylinder volume that no longer matches the hemisphere volume. Option 3: 12.2 cm implies a much larger base radius and inconsistent height for the required surface area ratio. Option 4: 14.0 cm simply repeats the original hemisphere radius, which would not satisfy the CSA:TSA ratio of 2 : 3 once recast. Option 5: 7.5 cm is too small and would require an excessively tall cylinder to match the hemisphere volume, which would not keep the desired surface area ratio.

Common Pitfalls:
Students often confuse volume conservation with surface area relationships and may try to equate areas instead of volumes. Another pitfall is to misinterpret the total surface area of a closed cylinder and forget to include both circular ends. Algebraic mistakes while handling the ratio CSA:TSA can also lead to incorrect expressions for h in terms of R. Finally, inaccurate approximations of the cube root of 3 or rounding too early can distort the final answer. Carefully following each formula and delaying approximations until the end helps avoid these errors.

Final Answer:
The radius of the base of the cylinder is approximately 9.7 cm.