Cones with equal diameters — relate curved surface areas via slant heights: Two cones have equal diameters (thus equal radii). If their slant heights are in the ratio 5 : 7, find the ratio of their curved surface areas.

Introduction / Context:The curved surface area (CSA) of a cone equals π r l. If two cones have the same radius, then CSA varies directly with slant height l. Therefore, the ratio of CSAs equals the ratio of slant heights.

Given Data / Assumptions:

• Equal diameters ⇒ r1 = r2.
• l1 : l2 = 5 : 7.

Concept / Approach:CSA1 : CSA2 = (π r l1) : (π r l2) = l1 : l2 when r is common. No additional computation is required.

Step-by-Step Solution:CSA ratio = l1 : l2 = 5 : 7

Verification / Alternative check:Pick r = 10, l1 = 5, l2 = 7: CSAs are 50π and 70π, giving 5 : 7.

Why Other Options Are Wrong:25 : 49 is the ratio of l^2 and would apply to area only if area depended on l^2 (it does not here since r is constant).

Common Pitfalls:Confusing cone CSA (π r l) with total surface area (π r l + π r^2), which would still keep a linear l dependence when r is common.

Final Answer:5 : 7