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A stone is dropped into a quiet lake and waves move in circles

Question

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

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A stone is dropped into a quiet lake and waves move in circles at the speed of

5 c m / s

. At the instant when the radius of the circular wave is

8 c m

, how fast is the enclosed area increasing?

8 mins ago

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8 mins ago

Text solutionVerified

The area of a circle

(A)

with radius

(r)

is given by

A = π r^{2}

Therefore, the rate of change of area

(A)

with respect to time

(t)

is given by,

\frac{d A}{d t} = \frac{d}{d t} (π r^{2}) = \frac{d}{d r} (π r^{2}) \frac{d r}{d t} = 2 π r \frac{d r}{d t},

[By chain rule]

It is given that

\frac{d r}{d t} = 5 c m

Thus, when

r = 8 c m,

\frac{d A}{d t} = 2 π (8) (5) = 80 π c m^{2} / s

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Question Text	A stone is dropped into a quiet lake and waves move in circles at the speed of $5 c m / s$ . At the instant when the radius of the circular wave is $8 c m$ , how fast is the enclosed area increasing?
Answer Type	Text solution:1
Upvotes	150