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 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
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.