A stone is dropped into a quiet lake and waves move in circles at the speed of 5 . At the instant when the radius of the circular wave is 8 , how fast is the enclosed area increasing?

Let at any instant of time t , the radius of the circular wave be r and the area enclosed be A , then A = …(i) Differentiating (i) w.r.t. t , we have dt = (2r) dt = 2 (8cm)(5 ) = 80 Therefore, rate of increase of enclosed area = 80 .

Application of Derivatives — Class 12 Maths Solution

ncert exercise SA NCERT Ex.6.1,Q.No. 5,Page 197

Question

A stone is dropped into a quiet lake and waves move in circles at the speed of $5{\rm{cm/s}}$. At the instant when the radius of the circular wave is $8{\rm{ cm}}$, how fast is the enclosed area increasing?

Step-by-step Solution

Let at any instant of time $t$, the radius of the circular wave be $r$ and the area enclosed be $A$, then $A = \pi {r^2}$

…(i)
Differentiating (i) w.r.t. $t$, we have
$\Rightarrow \cfrac{{dA}}{{dt}} = \pi (2r)\cfrac{{dr}}{{dt}} = 2\pi (8cm)(5{\rm{cm/sec}})$

$= 80\pi {\rm{c}}{{\rm{m}}^{\rm{2}}}{\rm{/sec}}$

Therefore, rate of increase of enclosed area $= 80\pi {\rm{c}}{{\rm{m}}^{\rm{2}}}{\rm{/sec}}$.

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