Would you like some cylinder in your sphere?
Calculus
Level
3
What is the maximum surface area for a right circular cylinder inscribed in a sphere of radius 10?
The answer is 1016.641.
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1 solution
Let half of the height be h. Let angle A=arcsin(h/R), where R is the radius of the sphere. V=2 3.1416(2sinAcosA+(cosA)^2)R^2 = 2 3.1416R^2(sin(2A)+(cos(2A)+1)/2) Differentiate regarding A, V'(A)=2cos(2A)-sin(2A)=0. Easily we confirm this critical point is a relative maximum, at tan(2A)=2. Hence we easily obtain cos(2A)=1/sqrt(5), and sin(2A)=2/sqrt(5) Plug back into V, V=(1+sqrt(5))3.1416*100 = 1016.6