Ex 6.5, 20 - Show that cylinder of given surface, max volume - Minima/ maxima (statement questions) - Geometry questions

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Ex 6.5,20 Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base. Let 𝑟 , ℎ be the Radius & Height of Cylinder respectively & 𝑉 , 𝑆 be the Volume & Surface area of Cylinder respectively Given Surface Area of Cylinder = 2𝜋 𝑟﷮2﷯+ 2𝜋𝑟ℎ S = 2𝜋 𝑟﷮2﷯+ 2𝜋𝑟ℎ S – 2𝜋 𝑟﷮2﷯= 2𝜋𝑟ℎ 𝑆 − 2𝜋 𝑟﷮2﷯﷮2𝜋𝑟﷯=ℎ ℎ= 𝑆 − 2𝜋 𝑟﷮2﷯﷮2𝜋𝑟﷯ Volume of Cylinder = 𝜋𝑟2ℎ V = 𝜋𝑟2ℎ We need to maximum volume Now, V = πr2h V = πr2 𝑆 − 2𝜋 𝑟﷮2﷯﷮2𝜋𝑟﷯﷯ V = 𝜋 𝑟﷮2﷯﷮2𝜋𝑟﷯ 𝑆 −2𝜋 𝑟﷮2﷯﷯ V = 𝑟﷮2﷯ 𝑆 −2𝜋 𝑟﷮2﷯﷯ V = 1﷮2﷯ 𝑆𝑟 −2𝜋 𝑟﷮3﷯﷯ Diff w.r.t 𝑟 𝑑𝑉﷮𝑑𝑟﷯= 1﷮2﷯ 𝑑 𝑠𝑟−2𝜋 𝑟﷮3﷯﷯﷮𝑑𝑟﷯ 𝑑𝑉﷮𝑑𝑟﷯= 1﷮2﷯ 𝑠−6𝜋 𝑟﷮2﷯﷯ Putting 𝑑𝑣﷮𝑑𝑟﷯=0 1﷮2﷯ 𝑠−6𝜋 𝑟﷮2﷯﷯=0 𝑠−6𝜋 𝑟﷮2﷯=0 Putting value of 𝑆 = 2𝜋𝑟2+ 2𝜋𝑟ℎ 2𝜋 𝑟﷮2﷯+2𝜋𝑟ℎ﷯−6𝜋 𝑟﷮2﷯=0 −4𝜋 𝑟2 + 2𝜋𝑟ℎ = 0 2𝜋𝑟ℎ −2𝑟+ℎ﷯=0 2𝜋𝑟 ℎ−2𝑟﷯=0 ℎ−2𝑟=0 ℎ=2𝑟 Finding 𝑑﷮2﷯𝑣﷮𝑑 𝑟﷮2﷯﷯ 𝑑𝑉﷮𝑑𝑟﷯= 1﷮2﷯ 𝑠−6𝜋 𝑟﷮2﷯﷯ 𝑑﷮2﷯𝑉﷮𝑑 𝑟﷮2﷯﷯= 1﷮2﷯ 𝑑 𝑆 − 6𝜋 𝑟﷮2﷯﷯﷮𝑑𝑟﷯ 𝑑﷮2﷯𝑣﷮𝑑 𝑟﷮2﷯﷯= 1﷮2﷯ 0−12𝜋𝑟﷯ 𝑑﷮2﷯𝑣﷮𝑑 𝑟﷮2﷯﷯=−6𝜋𝑟 < 0 ∴ 𝑑﷮2﷯𝑣﷮𝑑 𝑟﷮2﷯﷯<0 for ℎ=2𝑟 Hence volume of a cylinder is Maximum when 𝒉=𝟐𝒓

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