In a right circular cylinder, the ratio of curved surface area to total surface area is 1 : 2. If the total surface area is 616 cm^2, find the volume of the cylinder.

Introduction / Context:
We are given a relationship between curved surface area (CSA) and total surface area (TSA) for a cylinder and its TSA value. From the ratio, we deduce a relation between radius and height, then compute volume.

Given Data / Assumptions:

• CSA : TSA = 1 : 2.
• TSA = 616 cm^2.
• Standard formulas apply.

Concept / Approach:

• CSA = 2πrh.
• TSA = CSA + 2πr^2 = 2πrh + 2πr^2.
• Given TSA = 2 * CSA ⇒ 2πrh + 2πr^2 = 4πrh ⇒ r = h.
• With r = h, TSA = 4πr^2 ⇒ solve for r.
• Volume V = πr^2h = πr^3 with h = r.

Step-by-Step Solution:

Verification / Alternative check:
Compute CSA = 2πrh = 2*(22/7)*7*7 = 308 cm^2; TSA = 616 cm^2 satisfies the 1:2 ratio exactly.

Why Other Options Are Wrong:

• 1632, 1232, 1848 cm^3: Do not follow from the r = h constraint with the given TSA.
• None of these: Not applicable since 1078 cm^3 is exact.

Common Pitfalls:

• Misapplying the TSA formula (forgetting the two bases).
• Not reducing the ratio properly to r = h.

Final Answer:
1078 cm^3