Ex 6.1,5 - Chapter 6 Class 12 Application of Derivatives

Last updated at April 16, 2024 by Teachoo

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 ๐จ/๐ ๐ when r = 8 cm.
We know that
Area of circle = ฯr2
Now,
๐ ๐จ/๐ ๐ = (๐ (๐ ๐๐))/๐ ๐
๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ก
๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ก ร ๐๐/๐๐
๐๐ด/๐๐ก = ฯ (๐(๐2))/๐๐ ร ๐๐/๐๐ก
๐๐ด/๐๐ก = ฯ ร 2r ร ๐ ๐/๐ ๐
๐๐ด/๐๐ก = 2ฯr ร 5
๐๐ด/๐๐ก = 10ฯr
When r = 8 cm
โ ๐๐ด/๐๐กโค|_(๐ = 8) = 10 ร ฯ ร 8
โ ๐๐ด/๐๐กโค|_(๐ = 8) = 80ฯ
Since Area is in cm2 & time is in sec
โ ๐๐ด/๐๐กโค|_(๐ = 8)= 80ฯ cm2/sec
Hence Area is increasing at the rate of 80ฯ cm2/sec when r = 8 cm

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, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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