Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12


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


Ex 6.1, 8 A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of balloon increases when radius is 15 cm. Balloon is spherical Let r be the radius of spherical balloon . & v be the volume of spherical balloon. It is given that The balloon is inflated by pumping in in 900 cubic cm of gas per sec i.e. volume of balloon increasing at the rate of 900 cm3/sec i.e. ๐‘‘๐‘ฃ/๐‘‘๐‘ก = 900 cm3/sec We need to calculate the rate of change at which radius of Balloon increases when radius is 15 cm i.e. we need to calculate ๐‘‘๐‘Ÿ/๐‘‘๐‘ก when r = 15 cm we know that Volume of sphere = 4/3 ฯ€r3 i.e. V = 4/3 ฯ€r3 Differentiate w.r.t time(๐‘ก) ๐‘‘๐‘ฃ/๐‘‘๐‘ก = ๐‘‘(4/3 ๐œ‹๐‘Ÿ3)/๐‘‘๐‘ก ๐‘‘๐‘ฃ/๐‘‘๐‘ก = 4/3 ฯ€ (๐‘‘ (๐‘Ÿ3))/๐‘‘๐‘ก 900 = 4/3 ฯ€ . (๐‘‘(๐‘Ÿ3))/๐‘‘๐‘ก ร— ๐‘‘๐‘Ÿ/๐‘‘๐‘Ÿ 900 = 4/3 ฯ€ . (๐‘‘(๐‘Ÿ3))/๐‘‘๐‘ก ร— ๐‘‘๐‘Ÿ/๐‘‘๐‘ก 900 = 4/3 ฯ€ . 3r2 . ๐‘‘๐‘Ÿ/๐‘‘๐‘ก 900 = 4 ร— ฯ€ ร— r2 . ๐‘‘๐‘Ÿ/๐‘‘๐‘ก 900/(4 ร— ๐œ‹ ร—๐‘Ÿ2) = ๐‘‘๐‘Ÿ/๐‘‘๐‘ก ๐‘‘๐‘Ÿ/๐‘‘๐‘ก = (900 ร— 7)/(2 ร—22 ร—๐‘Ÿ2) We need to find ๐‘‘๐‘Ÿ/๐‘‘๐‘ก at r = 15 cm (From (1)) ๐‘‘๐‘Ÿ/๐‘‘๐‘ก = 900/(4 ร—๐œ‹ ร—(15)^2 ) ๐‘‘๐‘Ÿ/๐‘‘๐‘ก = 1/๐œ‹ Since radius in cm & time in sec So, ๐‘‘๐‘Ÿ/๐‘‘๐‘ก = ๐Ÿ/๐… ๐’„๐’Ž/๐’”๐’†๐’„ Hence ,the radius of the balloon is increasing at the rate of 1/๐œ‹ cm/sec when r = 15 cm.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.