Cone with known diameter and slant height — compute volume via height: A right circular cone has diameter 14 m (radius 7 m) and slant height 9 m. Find its volume (in m3).

Introduction / Context:To compute a cone’s volume when radius and slant height are given, first find the vertical height using Pythagoras on the right triangle formed by radius, height, and slant height. Then apply the standard cone volume formula.

Given Data / Assumptions:

• Radius r = 7 m (diameter 14 m).
• Slant height l = 9 m.
• Height h satisfies l^2 = r^2 + h^2.

Concept / Approach:Compute h = sqrt(l^2 − r^2). Then volume V = (1/3)π r^2 h. Leave the radical in simplest exact form for a clean expression.

Step-by-Step Solution:h = sqrt(9^2 − 7^2) = sqrt(81 − 49) = sqrt(32)V = (1/3)π * 7^2 * sqrt(32) = (1/3)π * 49 * sqrt(32)Therefore, V = 49 π √32 / 3 m3

Verification / Alternative check:Approximate: √32 ≈ 5.657; V ≈ (49 * 5.657 / 3)π ≈ (277.2/3)π ≈ 92.4π m3, consistent.

Why Other Options Are Wrong:All other forms mis-handle constants or the radical; the correct combination of r^2 and √32 is specific.

Common Pitfalls:Using l in place of h in the volume formula; forgetting to square the radius.

Final Answer:49 π √32/3 m3