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Showthat the height of a closed right circular cylinder of given surface andmaximum volume, is equal to the diameter of its base. |
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Answer» Surface area of cylinder ` A = 2pir(r+h)` `A = 2pir^2+2pirh` `h = (A-2pir^2)/(2pir)` Now, volume of cylinder `V = pir^2h` `=>V = pir^2((A-2pir^2)/(2pir))` `=>V = (Ar-2pir^3)/2` For maximum volume, `(dV)/(dr) = 0` `:. A/2-3pir^2 = 0` `=>3pir^2 = A/2` `=>6pir^2 = A->(1)` Now, As `A = 2pir^2+2pirh` `:. 2pir^2+2pirh = 6pir^2` `=>2pirh = 4pir^2` `=>h = 2r` So, height of given cylinder is double of the radius. In other words, height is equal to the diameter of the cylinder. |
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