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Question Number 115706 by Rio Michael last updated on 27/Sep/20
$$\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square},\:{A}\left({t}\right)\:\mathrm{is}\:\mathrm{increased} \\ $$$$\mathrm{at}\:\mathrm{a}\:\mathrm{rate}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{its}\:\mathrm{perimeter} \\ $$$$\mathrm{write}\:\mathrm{a}\:\mathrm{differential}\:\mathrm{equation}\:\mathrm{that}\:{A}\left({t}\right) \\ $$$$\mathrm{satisfy}\:,\:\mathrm{starting}\:\mathrm{from}\:\frac{{dA}}{{dt}}\:= \\ $$
Commented by Dwaipayan Shikari last updated on 27/Sep/20
$$\frac{\mathrm{dA}}{\mathrm{dt}}=\mathrm{p} \\ $$$$\frac{\mathrm{dA}}{\mathrm{dt}}=\mathrm{4}\sqrt{\mathrm{A}}\:\:\:\:\Rightarrow\frac{\mathrm{1}}{\mathrm{4}}\int\:\frac{\mathrm{dA}}{\:\sqrt{\mathrm{A}}}=\int\mathrm{dt} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{A}}\:=\mathrm{t}+\mathrm{C} \\ $$
Commented by Rio Michael last updated on 27/Sep/20
$$\mathrm{thanks} \\ $$
Commented by Dwaipayan Shikari last updated on 27/Sep/20
$$\mathrm{Amusing}\:,\mathrm{area}\:\mathrm{dependent}\:\mathrm{upon}\:\mathrm{time}.\:\mathrm{A}\:\mathrm{bacteria}\:\mathrm{probably} \\ $$
Answered by mr W last updated on 27/Sep/20
$${A}={a}^{\mathrm{2}} \\ $$$${P}=\mathrm{4}{a}=\mathrm{4}\sqrt{{A}} \\ $$$$\frac{{dA}}{{dt}}={cP}={k}\sqrt{{A}} \\ $$