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Example 3 - A stone is dropped into a quiet lake, waves move - Finding rate of change


  1. Chapter 6 Class 12 Application of Derivatives
  2. 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 4cm 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 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. ﷐𝑑𝑟﷮𝑑𝑡﷯ = 4 cm/sec. And we need to calculate how fast area increasing when waves is 10cm We need to calculate ﷐𝑑𝐴﷮𝑑𝑡﷯ at r = 10 We know that Area of circle is π r2 Now ﷐𝑑𝐴﷮𝑑𝑡﷯ = ﷐𝑑(𝜋𝑟2)﷮𝑑𝑡﷯ =𝜋﷐𝑑(𝑟2)﷮𝑑𝑡﷯ = π ﷐﷐𝑑﷐𝑟2﷯﷮𝑑𝑡﷯ × ﷐𝑑𝑟﷮𝑑𝑟﷯﷯ = π ﷐﷐𝑑𝑟2﷮𝑑𝑟﷯ × ﷐𝑑𝑟﷮𝑑𝑡﷯﷯ = π ﷐2𝑟 × ﷐𝒅𝒓﷮𝒅𝒕﷯﷯ = π ﷐2𝑟×𝟒﷯ = π [8r] Thus, ﷐𝑑𝐴﷮𝑑𝑡﷯ = π (8 r) When r = 10 ﷐﷐﷐𝑑𝐴﷮𝑑𝑡﷯﷯﷮𝑟 =10﷯ = π 8(10) ﷐﷐﷐𝑑𝐴﷮𝑑𝑡﷯﷯﷮𝑟 =10﷯ =80𝜋 Since Area is in cm2 & time is in sec ﷐𝑑𝐴﷮𝑑𝑡﷯ = 80𝜋 ﷐𝑐𝑚2﷮𝑠𝑒𝑐﷯ Hence the area is increasing at the rate of 80 π cm2/sec,

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.