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Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `2R/sqrt(3)` . Also find maximum volume. |
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Answer» `h^2=R^2-r^2` Volume of cylinder `r^2=R^2-h^2` `pir^2(2h)=2pir^2h` `2pi(R^2-h^2)h` `(dv)/(dh)=0` `d/dx[2pi(R^2-h^2)h]=0` `(R^2-h^2)+h*(-2h)=0` `R^2-3h^2=0` `h=R/sqrt3` Height of the cylinder at maximum `H=2h=(2R)/sqrt3` `V_(max)=2piR^2h=2pi(R^2-h^2)h` `=2pi(R^2-R^2/3)R/sqrt3` `=2pi(2R)^2/3*R/sqrt3` `(4piR^3)/(3sqrt3)` |
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