Cone – find slant height from volume and vertical height: A right circular cone has volume 100π cm^3 and vertical height 12 cm. Find its slant height l (in cm).

Introduction / Context:
Given a cone’s volume and height, we can determine its base radius r via V = (1/3)πr^2h. With r and h known, the slant height l follows from the Pythagorean relation l = √(r^2 + h^2). This integrates volume and geometry of cones.

Given Data / Assumptions:

• V = 100π cm^3
• h = 12 cm
• V = (1/3)πr^2h
• l = √(r^2 + h^2)

Concept / Approach:
Solve (1/3)πr^2h = 100π → cancel π → r^2 = 300 / h. Then compute r and finally l using the right-triangle relation with h as one leg and r as the other.

Step-by-Step Solution:
(1/3)πr^2 * 12 = 100π → 4r^2 = 100 → r^2 = 25 → r = 5 cml = √(r^2 + h^2) = √(25 + 144) = √169 = 13 cm

Verification / Alternative check:
Check volume with r = 5, h = 12: V = (1/3)π * 25 * 12 = (1/3)π * 300 = 100π cm^3, matching the given value exactly.

Why Other Options Are Wrong:
16 and 26 imply larger r or h; 9 is too small; 12 confuses vertical height with slant height. Only 13 cm satisfies both constraints simultaneously.

Common Pitfalls:
Forgetting to cancel π; using l = r + h incorrectly; rounding r before squaring; mixing slant and vertical height definitions.

Final Answer:
13 cm