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प्रश्न
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.
उत्तर
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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