Answer to Question #341049 in Calculus for boyyy

Question #341049

Find the angle of the largest right circular cone which can be inscribed in a sphere of

radius 9 inches.


1
Expert's answer
2022-05-17T14:43:28-0400


Let "R=" the radius of the sphere.

Let "AD=r, BD=h, \\angle ABC=\\theta."


"\\triangle ABC"

"AC=2r, AO=OC=R, \\angle AOC=2\\angle ABC=2\\theta"

The Law of Cosines



"(2r)^2=R^2+R^2-2R^2\\cos(2\\theta)""4r^2=2R^2(2\\sin^2\\theta)""r=R\\sin \\theta""h=r\\tan(\\theta\/2)=R\\sin \\theta\\tan(\\theta\/2)""V_{cone}=\\dfrac{1}{3}\\pi r^2 h""V_{cone}=V_{cone}(\\theta)=\\dfrac{1}{3}\\pi R^3\\sin^3 \\theta\\tan(\\theta\/2)""(V_{cone})'_{\\theta}=\\dfrac{1}{3}\\pi R^3\\bigg(3\\sin^2 \\theta\\cos(\\theta)\\tan(\\theta\/2)""+\\dfrac{1}{2}\\sin^3\\theta(\\dfrac{1}{\\cos ^2(\\theta\/2)})\\bigg)"

Find the critical numbr(s)



"(V_{cone})'_{\\theta}=0""\\dfrac{1}{3}\\pi R^3\\bigg(3\\sin^2 \\theta\\cos\\theta\\tan(\\theta\/2)""+\\dfrac{\\sin^3\\theta}{2\\cos ^2(\\theta\/2)}\\bigg)=0""\\dfrac{\\sin^3 \\theta(3\\cos\\theta+1)}{2\\cos ^2(\\theta\/2)}=0""\\cos \\theta=-\\dfrac{1}{3}"

If "0< \\theta<\\pi-\\cos^{-1}(1\/3), (V_{cone})'_{\\theta}>0, V_{cone}" increases.


If "\\pi-\\cos^{-1}(1\/3)<\\theta<\\pi, (V_{cone})'_{\\theta}<0, V_{cone}" decreases.


The volume of inscribed cone has the absolute maximum at


"\\theta=\\pi-\\cos^{-1}(1\/3)"

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