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
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Ex 6.1,5 A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing? Let r be the radius of circle & A be the Area of circle When stone is dropped into a lake waves move in a circle at the speed of 5 cm/sec Thus, ๐‘‘๐‘Ÿ/๐‘‘๐‘ก = 5cm/sec We need find how fast area increasing w. r. t time when radius is 8cm i.e. we need to find ๐‘‘๐ด/๐‘‘๐‘ก when r = 8 cm. We know that Area of circle = ฯ€r2 A = ฯ€r2 Different w. r. t time ๐‘‘๐ด/๐‘‘๐‘ก = (๐‘‘(๐œ‹๐‘Ÿ2))/๐‘‘๐‘ก ๐‘‘๐ด/๐‘‘๐‘ก = ฯ€ (๐‘‘(๐‘Ÿ2))/๐‘‘๐‘ก ๐‘‘๐ด/๐‘‘๐‘ก = ฯ€ (๐‘‘(๐‘Ÿ2))/๐‘‘๐‘ก ร— ๐‘‘๐‘Ÿ/๐‘‘๐‘Ÿ ๐‘‘๐ด/๐‘‘๐‘ก = ฯ€ (๐‘‘(๐‘Ÿ2))/๐‘‘๐‘Ÿ ร— ๐‘‘๐‘Ÿ/๐‘‘๐‘ก ๐‘‘๐ด/๐‘‘๐‘ก = ฯ€ ร— 2r. ๐‘‘๐‘Ÿ/๐‘‘๐‘ก ๐‘‘๐ด/๐‘‘๐‘ก = 2ฯ€r . ๐’…๐’“/๐’…๐’• ๐‘‘๐ด/๐‘‘๐‘ก = 2ฯ€r . 5 ๐‘‘๐ด/๐‘‘๐‘ก = 10ฯ€r When r = 8 cm โ”œ ๐‘‘๐ด/๐‘‘๐‘กโ”ค|_(๐‘Ÿ = 8) = 10 ร— ฯ€ ร— 8 โ”œ ๐‘‘๐ด/๐‘‘๐‘กโ”ค|_(๐‘Ÿ = 8) = 80ฯ€ Since Area is in cm2 & time is in sec โ”œ ๐‘‘๐ด/๐‘‘๐‘กโ”ค|_(๐‘Ÿ = 8)= 80ฯ€ ๐‘๐‘š2/๐‘ ๐‘’๐‘ (From (1): ๐‘‘๐‘Ÿ/๐‘‘๐‘ก=5) โ”œ ๐‘‘๐ด/๐‘‘๐‘กโ”ค|_(๐‘Ÿ = 8)= 80ฯ€ cm2/sec Hence Area is increasing at the rate of 80ฯ€ cm2/sec when r = 8 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.