Ex 6.1

Chapter 6 Class 12 Application of Derivatives
Serial order wise

Transcript

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))/๐๐ ๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ก ๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ก ร ๐๐/๐๐ ๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ ร ๐๐/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 ๐๐ด/๐๐กโค|_(๐ = 8) = 10 ร ฯ ร 8 โ ๐๐ด/๐๐กโค|_(๐ = 8) = 80ฯ Since Area is in cm2 & time is in sec โ ๐๐ด/๐๐กโค|_(๐ = 8)= 80ฯ cm2/sec Hence Area is increasing at the rate of 80ฯ cm2/sec when r = 8 cm