Finding rate of change
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Finding rate of change
Last updated at May 6, 2021 by Teachoo
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. π π/π π = 5 cm/sec We need find how fast area increasing when radius is 8 cm i.e. we need to find π π¨/π π when r = 8 cm. We know that Area of circle = Οr2 Now, π π¨/π π = (π (π ππ))/π π ππ΄/ππ‘ = Ο (π(π2))/ππ‘ ππ΄/ππ‘ = Ο (π(π2))/ππ‘ Γ ππ/ππ ππ΄/ππ‘ = Ο (π(π2))/ππ Γ ππ/ππ‘ ππ΄/ππ‘ = Ο Γ 2r Γ π π/π π ππ΄/ππ‘ = 2Οr Γ 5 ππ΄/ππ‘ = 10Οr When r = 8 cm β ππ΄/ππ‘β€|_(π = 8) = 10 Γ Ο Γ 8 β ππ΄/ππ‘β€|_(π = 8) = 80Ο Since Area is in cm2 & time is in sec β ππ΄/ππ‘β€|_(π = 8)= 80Ο cm2/sec Hence Area is increasing at the rate of 80Ο cm2/sec when r = 8 cm