Question

A cone has a diameter of 12 inches, a height of 14 inches, and a slant height 15 inches. What is the height of the right circular cone?

Answer 1

Those three numbers are inconsistent for a right circular cone. If the diameter is 12, the radius is r = 6, and for a right cone the slant height l, radius r, and height h satisfy l^2 = r^2 + h^2. With l = 15 and r = 6,

h = sqrt(15^2 − 6^2) = sqrt(225 − 36) = sqrt(189) = 3√21 ≈ 13.747.

So a right circular cone with diameter 12 and slant height 15 has height 3√21 (not 14). If you meant something else, please clarify.