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Question
If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.
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Solution
Let, the radius of a circle be r .
We have, C = 2πr and let `(dC)/dt` = k ...(i)
Now, A = πr2
`(dA)/dt = 2πr (dr)/dt` ...(ii)
and `(dC)/dt = 2π (dr)/dt`
k = `2π (dr)/dt` ...[From (i)]
`\implies (dr)/dt = k/(2π)` ...(iii)
Put the value of `(dr)/dt` from equation (iii) in (ii)
`\implies (dA)/dt = 2πr xx k/(2π)` = kr
`implies (dA)/dt ∝ r`
Hence Proved.
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