JEEOlympiad
A right circular cone is inscribed in a sphere of radius R so that…
Question
A right circular cone is inscribed in a sphere of radius R so that its apex and base circle lie on the sphere. The maximum possible volume of the cone equals (m/n) times the volume of the sphere, where m/n is in lowest terms. Find m + n.
✓ Verified answer: 35checked by our engine — not a guess
Step-by-step solution
Let the cone have height h measured from apex, with the apex and base on the sphere.
If the base is at distance (h - R) from the center along the axis, the base radius r satisfies r^2 = R^2 - (h - R)^2 = h(2R - h).
The volume is V = (1/3)π r^2 h = (1/3)π h^2 (2R - h).
Then dV/dh = (1/3)π (4Rh - 3h^2) = 0 gives h = 4R/3.
So r^2 = (4R/3)(2R - 4R/3) = (4R/3)(2R/3) = 8R^2/9, and V_max = (1/3)π (8R^2/9)(4R/3) = 32π R^3/81.
The sphere volume is (4/3)π R^3, so the ratio is (32/81)/(4/3) = 32/(81) · 3/4 = 8/27.
Thus m/n = 8/27 and m + n = 35.
Final answer35
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