Sphere vs hemisphere with equal total surface area — compare volumes: A sphere and a hemisphere have the same total surface area (hemisphere area includes its circular base). Find the ratio of their volumes (sphere : hemisphere).

Introduction / Context:
Be careful with the hemisphere’s surface area: “total” means curved area plus the flat circular base. Setting the total areas equal lets us link their radii and then compare volumes.

Given Data / Assumptions:

• Sphere: S_s = 4πR^2, V_s = (4/3)πR^3.
• Hemisphere (including base): S_h = 2πr^2 + πr^2 = 3πr^2, V_h = (2/3)πr^3.
• Equal total surface areas: 4πR^2 = 3πr^2.

Concept / Approach:
From 4πR^2 = 3πr^2, deduce r^2 = (4/3)R^2 ⇒ r = (2/√3)R. Substitute into the volume expressions and simplify the ratio V_s : V_h.

Step-by-Step Solution:
Set 4πR^2 = 3πr^2 ⇒ r = (2/√3)RV_s : V_h = [(4/3)πR^3] : [(2/3)πr^3] = 2 * (R^3 / r^3)R/r = √3/2 ⇒ (R/r)^3 = (√3/2)^3 = 3√3/8Thus V_s : V_h = 2 * (3√3/8) = 3√3/4 : 1

Verification / Alternative check:
Pick R = 2 and compute r = 2/√3; substituting numerically reproduces the symbolic ratio.

Why Other Options Are Wrong:
They drop or misplace the factor 2 or the cube when converting area equality into a volume ratio.

Common Pitfalls:
Using hemisphere curved area only (2πr^2) instead of total (3πr^2).

Final Answer:
3√3/4 : 1