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

Ex 6.1, 9 - A balloon has a variable radius. Find rate - Finding rate of change


  1. Chapter 6 Class 12 Application of Derivatives
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Ex 6.1,9 A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm. Balloon is spherical let r be the radius of balloon . & v be the volume of balloon. We need to find. rate at which balloon volume is increasing with the radius when r is 10cm i.e. we need to find change of volume w.r.t radius when r = 10 i.e. we need to find ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ when r = 10 we know that volume of sphere = 4๏ทฎ3๏ทฏ ฯ€r3 v = 4๏ทฎ3๏ทฏ ฯ€r3 Differentiate w. r. t radius ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ = ๐‘‘ 4๏ทฎ3๏ทฏ ๐œ‹๐‘Ÿ3๏ทฏ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ = 4๏ทฎ3๏ทฏ ฯ€ ๐‘‘ ๐‘Ÿ3๏ทฏ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ = 4๏ทฎ3๏ทฏ ฯ€ 3 ๐‘Ÿ๏ทฎ2๏ทฏ ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ = 4๐œ‹ ๐‘Ÿ๏ทฎ2๏ทฏ When r = 10 ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ = 4 ร— ฯ€ ร— (10)2 ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ = 400 ฯ€ Since volume is in cm3 & Radius is in cm So, ๐‘‘๐‘ฃ๏ทฎ๐‘‘๐‘Ÿ๏ทฏ = 400ฯ€ ๐’„๐’Ž๐Ÿ‘๏ทฎ๐’„๐’Ž๏ทฏ Hence, volume is increasing at the rate of 400 ฯ€ cm3/cm when r = 10 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.