Example 3 - A stone is dropped into a quiet lake, waves move

Example 3 - Chapter 6 Class 12 Application of Derivatives - Part 2
Example 3 - Chapter 6 Class 12 Application of Derivatives - Part 3

  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

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 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.