1. Class 12
2. Important Question for exams Class 12

Transcript

Example 38 Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone. Let OC = r be the radius of cone & OA = h be height of cone & ∠ OAQ = α be the semi−vertical angle of cone Let OE = x be the radius of cylinder Height of cylinder = OO’ From (1) & (2) ﷐𝑥﷮ℎ − ﷐𝑂﷮′﷯𝑂﷯=﷐𝑟﷮ℎ﷯ ﷐ℎ𝑥﷮𝑟﷯=ℎ−﷐𝑂﷮′﷯𝑂 ﷐𝑂﷮′﷯𝑂=ℎ−﷐ℎ𝑥﷮𝑟﷯ ﷐𝑂﷮′﷯𝑂= ﷐ℎ𝑟 − ℎ𝑥﷮𝑟﷯ ﷐𝑂﷮′﷯𝑂= ﷐ℎ﷐𝑟 − 𝑥﷯﷮𝑟﷯ Now, Curved Surface Area Of cylinder = 2𝜋× Radius of cylinder × Height of cylinder S = 2𝜋 × 𝑥 × ﷐𝑂﷮′﷯𝑂 S = 2𝜋𝑥﷐ℎ﷐𝑟 − 𝑥﷯﷮𝑟﷯ S = ﷐2𝜋ℎ﷮𝑟﷯﷐𝑟𝑥−﷐𝑥﷮2﷯﷯ S = 𝑘﷐𝑟𝑥−﷐𝑥﷮2﷯﷯ We need to minimize S, So, finding S’(x) S’ = ﷐𝑑﷐𝑘﷐𝑟𝑥 − ﷐𝑥﷮2﷯﷯﷯﷮𝑑𝑥﷯ S’ = 𝑘 ﷐𝑑﷐𝑟𝑥 − ﷐𝑥﷮2﷯﷯﷮𝑑𝑥﷯ S’ = 𝑘﷐𝑟−2𝑥﷯ Putting S’ = 0 0 = 𝑘﷐𝑟−2𝑥﷯ 𝑟−2𝑥 = 0 𝑥 = ﷐𝑟﷮2﷯ Now, Finding S’’(x) at x = ﷐𝑟﷮2﷯ S’’ = ﷐𝑑﷐𝑘﷐𝑟 − 2𝑥﷯﷯﷮𝑑𝑥﷯ S’’ = 𝑘 ﷐𝑑﷐𝑟 − 2𝑥﷯﷮𝑑𝑥﷯ S’’ = 𝑘 ﷐0−2﷯ S’’ = −2𝑘 So, S’’ = −2𝑘 Hence, at 𝑥=﷐𝑟﷮2﷯ ﷐﷐𝑆′′﷮𝑥 = ﷐𝑟﷮2﷯﷯﷯<0 ∴ 𝒙=﷐𝒓﷮𝟐﷯ is maxima of S. Hence, radius of cylinder with greatest curved surface area which can be inscribed in a given cone is half of that cone.

Class 12
Important Question for exams Class 12