Cone with r : h = 5 : 12 and known volume — find slant height: A right circular cone has radius : height = 5 : 12 and volume 314 2/7 cm3. Find its slant height (in centimeters).

Introduction / Context:
When a cone’s radius and height are in a fixed ratio, we can parameterize them with a common factor and use the volume to determine that factor. The slant height then follows from Pythagoras.

Given Data / Assumptions:

• r : h = 5 : 12 ⇒ r = 5k, h = 12k for some k > 0.
• V = 314 2/7 cm3 = 2200/7 cm3 (since 314*7 + 2 = 2200).
• Use π = 22/7 (implied by mixed fraction format), typical in aptitude problems.

Concept / Approach:
Compute V in terms of k: V = (1/3)π r^2 h = (1/3)π * (25k^2) * (12k) = 100π k^3. Equate to 2200/7 to obtain k, then compute slant height l = sqrt(r^2 + h^2).

Step-by-Step Solution:
100π k^3 = 2200/7 with π = 22/7 ⇒ 100 * (22/7) * k^3 = 2200/7Cancel (22/7): 100k^3 = 100 ⇒ k = 1Thus r = 5 cm, h = 12 cml = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 cm

Verification / Alternative check:
Plug r = 5, h = 12 back into V: (1/3)*π*25*12 = 100π = 100*(22/7) = 2200/7 cm3, matching the data.

Why Other Options Are Wrong:
15, 16, 18 cm do not satisfy l^2 = r^2 + h^2 with r : h = 5 : 12.

Common Pitfalls:
Mistaking the mixed fraction as 314.27; forgetting to square r in the volume formula.

Final Answer:
13 cm