Circle — The area increases by 22 sq cm when the radius increases by 1 cm. Find the original radius.

Introduction / Context:
The change in area of a circle due to a change in radius can be written directly. If radius goes from r to r + 1, the area change is π[(r + 1)^2 − r^2] = π(2r + 1). This gives a linear equation in r.

Given Data / Assumptions:

• ΔA = 22 sq cm
• Radius increment = 1 cm
• Use π ≈ 22/7 (makes an exact integer solution here)

Concept / Approach:
Set π(2r + 1) = 22 and solve for r.

Step-by-Step Solution:
π(2r + 1) = 22 ⇒ (22/7)(2r + 1) = 222r + 1 = 22 * (7/22) = 7 ⇒ 2r = 6 ⇒ r = 3 cm

Verification / Alternative check:
Compute areas: A(r) = πr^2 = (22/7)*9 = 198/7; A(r + 1) = (22/7)*16 = 352/7; difference = 154/7 = 22 ✔

Why Other Options Are Wrong:
3.5, 3.2, 4, and 6 cm do not satisfy π(2r + 1) = 22.

Common Pitfalls:
Using circumference difference formulas instead of area, or forgetting (a + b)^2 − a^2 = 2ab + b^2.

Final Answer:
3 cm