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Ex 6.1, 5 - A stone is dropped into a quiet lake and waves move

Ex 6.1,5 - Chapter 6 Class 12 Application of Derivatives - Part 2
Ex 6.1,5 - Chapter 6 Class 12 Application of Derivatives - Part 3

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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 Given that When stone is dropped into a lake waves move in a circle at speed of 5 cm/sec i.e. Radius of circle increasing at a rate of 4 cm / sec. i.e. ๐’…๐’“/๐’…๐’• = 5 cm/sec We need find how fast area increasing when radius is 8 cm i.e. we need to find ๐’…๐‘จ/๐’…๐’• when r = 8 cm. We know that Area of circle = ฯ€r2 Now, ๐’…๐‘จ/๐’…๐’• = (๐’…(๐…๐’“๐Ÿ))/๐’…๐’• 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 ร— ๐’…๐’“/๐’…๐’• ๐‘‘๐ด/๐‘‘๐‘ก = 2ฯ€r ร— 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€ cm2/sec Hence Area is increasing at the rate of 80ฯ€ cm2/sec when r = 8 cm

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.