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  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
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Misc 18 Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one-third that of the cone and the greatest volume of cylinder is ﷐4﷮27﷯ 𝜋ℎ3 tan2 𝛼 Let PQR be the cone of height h i.e. PO = h & α be the semi vertical angle of cone Let 𝑥 be the radius of cylinder ABCD which is inscribed in the cone PQR Now, Height of cylinder = OO’ = PO – PO’ In ∆AP𝑂 tan α = ﷐𝐴﷐𝑂﷮′﷯﷮𝑃﷐𝑂﷮′﷯﷯ tan α = ﷐𝑥﷮𝑉﷐𝑂﷮′﷯﷯ VO’ = ﷐𝑥﷮﷐tan﷮α ﷯﷯ VO’ = 𝑥 cotα Now Height of cylinder = OO’ OO’ = VO – VO’ OO’ = h – 𝑥 cot α We need to maximize volume of cylinder Let V be the volume of cylinder V = π ﷐﷐𝑟𝑎𝑑𝑖𝑢𝑠 ﷯﷮2﷯﷐ℎ𝑒𝑖𝑔ℎ𝑡﷯ V = π ﷐﷐﷐𝐴﷮′﷯﷐𝑂﷮′﷯﷯﷮2﷯﷐𝑂 𝑂′﷯ V = π ﷐﷐𝑥﷯﷮2﷯﷐ℎ−𝑥﷐cot﷮α﷯﷯ V = π 𝑥2 ﷐ℎ−𝑥α﷯ V = π h𝑥2 – π cot α. 𝑥3 Differentiating w.r.t 𝑥 ﷐𝑑𝑉﷮𝑑𝑥﷯=﷐𝑑﷐𝜋ℎ﷐𝑥﷮2﷯−𝜋﷐cot﷮α.﷐𝑥﷮3﷯﷯﷯﷮𝑑𝑥﷯ ﷐𝑑𝑉﷮𝑑𝑥﷯= π h﷐𝑑﷐﷐𝑥﷯﷮2﷯﷮𝑑𝑥﷯−𝜋﷐cot﷮α.﷐𝑑﷐﷐𝑥﷯﷮3﷯﷮𝑑𝑥﷯﷯ ﷐𝑑𝑉﷮𝑑𝑥﷯= πh. 2𝑥 – π cot α. 3𝑥2 ﷐𝑑𝑉﷮𝑑𝑥﷯= 2πh𝑥 – 3π cot α 𝑥2 Putting ﷐𝑑𝑉﷮𝑑𝑥﷯= 0 2π h 𝑥 – 3π cot α 𝑥2 = 0 3π cot α 𝑥2 = 2π h 𝑥 𝑥 = ﷐2𝜋ℎ 𝑥﷮3𝜋﷐cot﷮ α.𝑥﷯﷯ 𝑥 = ﷐2ℎ﷮3﷐cot﷮ α﷯﷯ 𝑥 = ﷐2ℎ﷮3﷯﷐tan﷮α﷯ Now finding ﷐﷐𝑑﷮2﷯𝑉﷮𝑑﷐𝑥﷮2﷯﷯ ﷐﷐𝑑﷮2﷯𝑉﷮𝑑﷐𝑥﷮2﷯﷯= ﷐𝑑﷐2𝜋 ℎ𝑥 − 3𝜋 𝑐𝑜𝑡α . ﷐ 𝑥﷮2﷯﷯﷮𝑑𝑥﷯ ﷐﷐𝑑﷮2﷯𝑉﷮𝑑𝑥﷯= 2π h – 3π cot α . 2𝑥 ﷐﷐𝑑﷮2﷯𝑣﷮𝑑﷐𝑥﷮2﷯﷯=2π h – 6π cot α . 𝑥 Putting value of 𝑥 = ﷐2ℎ﷮3﷐𝑐𝑜𝑡﷮α﷯﷯ ﷐﷐𝑑﷮2﷯𝑣﷮𝑑﷐𝑥﷮2﷯﷯=2𝜋ℎ−4𝜋ℎ = –2π h < 0 ⇒ ﷐﷐𝑑﷮2﷯𝑣﷮𝑑﷐𝑥﷮2﷯﷯ < 0 at 𝑥 = ﷐2ℎ﷮3﷐cot﷮ α﷯﷯ ⇒ Hence, V is maximum when 𝑥 = ﷐2ℎ﷮3﷐cot﷮α﷯﷯ From (1) OO’ = h – 𝑥 cot α OO’ = h – ﷐2ℎ﷮3﷐cot﷮α ﷯﷯ ×α OO’ = h – ﷐2ℎ﷮3﷯ OO’ = ﷐ℎ﷮ 3﷯ Thus, Volume is maximum when Height of cylinder is = ﷐1﷮3﷯ ×ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑛𝑒 Finding Maximum Volume V = π 𝑥2 ﷐ℎ −𝑥﷐cot﷮α﷯﷯ V = π ﷐﷐﷐2ℎ﷮3﷐cot﷮α﷯﷯﷯﷮2﷯﷐ℎ−﷐2ℎ﷮3﷐cot﷮α﷯﷯ ×﷐cot﷮α﷯﷯ V = π ﷐﷐4﷐ℎ﷮2﷯﷮9﷐﷐ cot﷮2﷯﷮α﷯﷯﷯﷐ℎ−﷐2ℎ﷮3﷯﷯ V = π ﷐﷐4﷐ℎ﷮2﷯﷮9﷐﷐cot﷮2﷯﷮α﷯﷯﷯﷐﷐ℎ﷮3﷯﷯ V = ﷐4 ﷮9 ×3﷯﷐﷐𝜋﷐ℎ﷮2﷯ ×ℎ﷮﷐﷐cot﷮2﷯﷮α﷯﷯﷯ V = ﷐4﷮27﷯﷐𝜋﷐ℎ﷮3﷯﷐𝜋﷐ℎ﷮3﷯﷮﷐﷐cot﷮2﷯﷮α﷯﷯﷯ V = ﷐4﷮2𝜋﷯𝜋﷐ℎ﷮3﷯.﷐﷐tan﷮2﷯﷮α﷯ Hence height of cylinder is one third of cone & greatest volume of cylinder is ﷐𝟒﷮𝟐𝝅﷯𝝅﷐𝒉﷮𝟑﷯﷐﷐𝐭𝐚𝐧﷮𝟐﷯﷮𝜶﷯

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here