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. 𝒅𝒓/𝒅𝒕 = 5 cm/sec
We need find how fast area increasing when radius is 8 cm
i.e. we need to find 𝒅𝑨/𝒅𝒕 when r = 8 cm.
We know that
Area of circle = πr2
Now,
𝒅𝑨/𝒅𝒕 = (𝒅(𝝅𝒓𝟐))/𝒅𝒕
𝑑𝐴/𝑑𝑡 = π (𝑑(𝑟2))/𝑑𝑡
𝑑𝐴/𝑑𝑡 = π (𝑑(𝑟2))/𝑑𝑡 × 𝑑𝑟/𝑑𝑟
𝑑𝐴/𝑑𝑡 = π (𝑑(𝑟2))/𝑑𝑟 × 𝑑𝑟/𝑑𝑡
𝑑𝐴/𝑑𝑡 = π × 2r × 𝒅𝒓/𝒅𝒕
𝑑𝐴/𝑑𝑡 = 2πr × 5
𝑑𝐴/𝑑𝑡 = 10πr
When r = 8 cm
├ 𝑑𝐴/𝑑𝑡┤|_(𝑟 = 8) = 10 × π × 8
├ 𝑑𝐴/𝑑𝑡┤|_(𝑟 = 8) = 80π
Since Area is in cm2 & time is in sec
├ 𝑑𝐴/𝑑𝑡┤|_(𝑟 = 8)= 80π cm2/sec
Hence Area is increasing at the rate of 80π 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!