Example 3 - Chapter 6 Class 12 Application of Derivatives

Last updated at April 16, 2024 by

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Example 3 A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second. At the instant, when the radius of the circular wave is 10 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 dropped into a lake wavers movie in circle at speed to 4 cm per sec. i.e. Radius of circle increasing at a rate of 4 cm / sec. i.e. ๐’…๐’“/๐’…๐’• = 4 cm/sec We need to calculate how fast area increasing when waves is 10 cm i.e. we need to calculate ๐’…๐‘จ/๐’…๐’• at r = 10 We know that Area of circle is ฯ€r2 Now ๐‘‘๐ด/๐‘‘๐‘ก = (๐‘‘(๐œ‹๐‘Ÿ2))/๐‘‘๐‘ก =๐œ‹ (๐‘‘(๐‘Ÿ2))/๐‘‘๐‘ก = ฯ€ [๐‘‘(๐‘Ÿ2)/๐‘‘๐‘ก ร— ๐‘‘๐‘Ÿ/๐‘‘๐‘Ÿ] = ฯ€ [๐‘‘๐‘Ÿ2/๐‘‘๐‘Ÿ ร— ๐‘‘๐‘Ÿ/๐‘‘๐‘ก] = ฯ€ [2๐‘Ÿ ร— ๐’…๐’“/๐’…๐’•] = ฯ€ [2๐‘Ÿ ร— ๐Ÿ’] = 8ฯ€r When r = 10 โ”œ ๐‘‘๐ด/๐‘‘๐‘กโ”ค|_(๐‘Ÿ =10) = 8ฯ€ ร— 10 โ”œ ๐‘‘๐ด/๐‘‘๐‘กโ”ค|_(๐‘Ÿ =10) =80๐œ‹ Since Area is in cm2 & time is in sec ๐‘‘๐ด/๐‘‘๐‘ก = 80๐… ๐’„๐’Ž๐Ÿ/๐’”๐’†๐’„ Hence the area is increasing at the rate of 80 ฯ€ cm2/sec,

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo