Question

A stone is dropped into a quiet lake and waves move in circles at the speed of 5cm/s. At the instant when the radius of the circular wave is 8cm, how fast is the enclosed area increasing? (Class 12 Physics)

Answer 1

Answer:

80π cm²/s

Explanation:

given:

rate of change of radius=dr/dt=5 cm/s

to find:

rate of change of area when r=8 cm

we know,

area of circle (A)=πr²

so,rate of change of area=dA/dt

=d(πr²)/dt

=2πr*d(r)/dt

=2πr*5

so,when r=8cm,then rate of change of area=2π*8*5 cm²/s

=80π cm²/s

Answer 2

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We know that the area of a circle with radius “r” is given by A = πr2.

Hence, the rate of change of area “A’ with respect to the time “t” is given by:

dA/dt = (d/dt) πr2

By using the chain rule, we get:

(d/dr)(πr2). (dr/dt) = 2πr.(dr/dt)

It is given that, dr/dt = 4 cm/sec

Therefore, when r = 10 cm,

dA/dt = 2π. (10). (4)

dA.dt = 80 π

Hence,

when r = 10 cm, the enclosing area is increasing at a rate of 80π cm2/sec

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Hope It's Helpful....:)