Examples

Chapter 6 Class 12 Application of Derivatives
Serial order wise

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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๐2))/๐๐ก =๐ (๐(๐2))/๐๐ก = ฯ [๐(๐2)/๐๐ก ร ๐๐/๐๐] = ฯ [๐๐2/๐๐ ร ๐๐/๐๐ก] = ฯ [2๐ ร ๐๐/๐๐] = ฯ [2๐ ร ๐] = 8ฯr When r = 10 โ ๐๐ด/๐๐กโค|_(๐ =10) = 8ฯ ร 10 โ ๐๐ด/๐๐กโค|_(๐ =10) =80๐ Since Area is in cm2 & time is in sec ๐๐ด/๐๐ก = 80๐ ๐๐๐/๐๐๐ Hence the area is increasing at the rate of 80 ฯ cm2/sec,