Ex 6.1, 12 - Radius of an air bubble is increasing at 1/2 cm/s

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Ex 6.1,12 - Chapter 6 Class 12 Application of Derivatives - Part 2

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Ex 6.1,12 - Chapter 6 Class 12 Application of Derivatives - Part 3 Ex 6.1,12 - Chapter 6 Class 12 Application of Derivatives - Part 4

  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Ex 6.1, 12 The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?Since Air Bubble is spherical Let r be the radius of bubble & V be the volume of bubble Given that Radius of an air bubble is increasing at the rate of 1/2 cm/s i.e. ๐’…๐’“/๐’…๐’• = ๐Ÿ/๐Ÿ cm/sec We need to calculate the rate is the volume of the bubble increasing when the radius is 1 cm i.e. we need to calculate ๐’…๐‘ฝ/๐’…๐’• when r = 1 cm We know that Volume of sphere = V = ๐Ÿ’/๐Ÿ‘ ฯ€r3 Now, ๐‘‘๐‘‰/๐‘‘๐‘ก = ๐‘‘(4/3 ๐œ‹๐‘Ÿ3)/๐‘‘๐‘ก ๐‘‘๐‘‰/๐‘‘๐‘ก = 4/3 ฯ€ (๐‘‘ (๐‘Ÿ3))/๐‘‘๐‘ก ๐‘‘๐‘‰/๐‘‘๐‘ก = 4/3 ฯ€ (๐‘‘ (๐‘Ÿ3))/๐‘‘๐‘ก ๐‘‘๐‘‰/๐‘‘๐‘ก = 4/3 ฯ€ . (๐‘‘(๐‘Ÿ3))/๐‘‘๐‘ก ร— ๐’…๐’“/๐’…๐’“ ๐‘‘๐‘‰/๐‘‘๐‘ก = 4/3 ฯ€ . (๐‘‘(๐‘Ÿ3))/๐‘‘๐‘ก ร— ๐‘‘๐‘Ÿ/๐‘‘๐‘ก ๐‘‘๐‘‰/๐‘‘๐‘ก = 4/3 ฯ€ .3r2 . ๐‘‘๐‘Ÿ/๐‘‘๐‘ก ๐‘‘๐‘‰/๐‘‘๐‘ก = 4/3 ฯ€ . 3r2 ร— 1/2 ๐‘‘๐‘‰/๐‘‘๐‘ก = 2๐œ‹๐‘Ÿ^2 We need to find ๐‘‘๐‘‰/๐‘‘๐‘ก at r = 1 cm ๐‘‘๐‘‰/๐‘‘๐‘ก = 2๐œ‹ใ€–(1)ใ€—^2 ("From (1): " ๐‘‘๐‘Ÿ/๐‘‘๐‘ก=1/2 cm/s) ๐’…๐‘ฝ/๐’…๐’• = ๐Ÿ๐… Since Volume is in cm3 & time is in sec โˆด ๐‘‘๐‘‰/๐‘‘๐‘ก = ๐Ÿ๐… cm3/sec Hence, Volume is increasing at rate of 2๐œ‹ cm3/sec

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.