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Last updated at Jan. 7, 2020 by Teachoo

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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, ๐๐/๐๐ก = 5cm/sec We need find how fast area increasing w. r. t time when radius is 8cm i.e. we need to find ๐๐ด/๐๐ก when r = 8 cm. We know that Area of circle = ฯr2 A = ฯr2 Different w. r. t time ๐๐ด/๐๐ก = (๐(๐๐2))/๐๐ก ๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ก ๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ก ร ๐๐/๐๐ ๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ ร ๐๐/๐๐ก ๐๐ด/๐๐ก = ฯ ร 2r. ๐๐/๐๐ก ๐๐ด/๐๐ก = 2ฯr . ๐ ๐/๐ ๐ ๐๐ด/๐๐ก = 2ฯr . 5 ๐๐ด/๐๐ก = 10ฯr When r = 8 cm โ ๐๐ด/๐๐กโค|_(๐ = 8) = 10 ร ฯ ร 8 โ ๐๐ด/๐๐กโค|_(๐ = 8) = 80ฯ Since Area is in cm2 & time is in sec โ ๐๐ด/๐๐กโค|_(๐ = 8)= 80ฯ ๐๐2/๐ ๐๐ (From (1): ๐๐/๐๐ก=5) โ ๐๐ด/๐๐กโค|_(๐ = 8)= 80ฯ cm2/sec Hence Area is increasing at the rate of 80ฯ cm2/sec when r = 8 cm

Chapter 6 Class 12 Application of Derivatives

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About the Author

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.