A solid sphere and a solid hemisphere have the same total surface area (curved surface plus base for the hemisphere). If π is taken as 22/7, what is the ratio of the volume of the sphere to the volume of the hemisphere?

Introduction / Context:
This problem compares a solid sphere and a solid hemisphere that share the same total surface area. You are asked to find the ratio of their volumes. The question checks how well you know the formulas for surface area and volume of spheres and hemispheres and whether you can manipulate ratios without needing actual numerical radii.

Given Data / Assumptions:
• Total surface area of the sphere equals the total surface area of the hemisphere.
• Total surface area of a sphere of radius R is 4πR2.
• Total surface area of a hemisphere of radius r is 3πr2 (curved surface 2πr2 plus base πr2).
• Volume of a sphere of radius R is (4/3)πR3.
• Volume of a hemisphere of radius r is (2/3)πr3.

Concept / Approach:
First, equate the total surface areas to obtain a relationship between the radii of the sphere and hemisphere. Then express the volume of each solid in terms of its radius. Using the radius ratio found from surface areas, you can compute the ratio of the volumes. All constants like π will cancel in the ratio, so the calculation simplifies nicely.

Step-by-Step Solution:
Step 1: Let the sphere have radius Rs and the hemisphere have radius Rh. Step 2: Equate surface areas: 4πRs2 = 3πRh2. Step 3: Cancel π: 4Rs2 = 3Rh2, so Rs2 / Rh2 = 3 / 4. Step 4: Take square roots: Rs / Rh = √3 / 2. Step 5: Volume of sphere: Vs = (4/3)πRs3. Volume of hemisphere: Vh = (2/3)πRh3. Step 6: Ratio Vs : Vh = [(4/3)πRs3] / [(2/3)πRh3] = 2 × (Rs3 / Rh3). Step 7: Substitute Rs / Rh = √3 / 2. Then Rs3 / Rh3 = (√3 / 2)3 = (3√3) / 8. Step 8: So Vs : Vh = 2 × (3√3 / 8) = 3√3 / 4, that is 3√3 : 4.

Verification / Alternative check:
You can pick convenient radii that satisfy Rs / Rh = √3 / 2, for example Rh = 2 and Rs = √3. Compute numerical surface areas and volumes with π = 22/7 to confirm the equality of surface areas and the derived volume ratio. The numerical results will be proportional and match 3√3 : 4 after simplification.

Why Other Options Are Wrong:
• 4 : 3√3 inverts the correct ratio and would result from mistakenly taking hemisphere volume over sphere volume.
• 3 : 4√3 arises from incorrectly squaring rather than cubing the radius ratio.
• 1 : 12√3 does not follow from any correct manipulation of the formulas and is far from the simplified expression.

Common Pitfalls:
A common error is to forget that hemisphere total surface area includes the circular base. Another pitfall is mishandling powers when converting a radius ratio into a volume ratio, since volume depends on the cube of the radius. Careful step management and simplification of ratios avoid these mistakes.

Final Answer:
The ratio of the sphere volume to the hemisphere volume is 3√3 : 4.