Question Number 3330 by Filup last updated on 11/Dec/15
$$\mathrm{If}\:\mathrm{you}\:\mathrm{had}\:\mathrm{a}\:\mathrm{rectangle}\:\mathrm{with}\:\mathrm{side}\:\mathrm{lengths} \\ $$$${h}\:\mathrm{and}\:{b},\:\mathrm{the}\:\mathrm{area}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}: \\ $$$${A}\left({h},\:{b}\right)={hb} \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{I}\:\mathrm{were}\:\mathrm{to}\:\mathrm{change}\:\left(\mathrm{increase}\:\mathrm{or}\:\mathrm{decrease}\right) \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{one}\:\mathrm{or}\:\mathrm{both}\:\mathrm{of}\:\mathrm{the}\:\mathrm{variables} \\ $$$$\mathrm{in}\:{A},\:\mathrm{how}\:\mathrm{can}\:\mathrm{I}\:\mathrm{write}\:\mathrm{the}\:\mathrm{change}\:\mathrm{in}\:\mathrm{area}? \\ $$
Answered by Rasheed Soomro last updated on 11/Dec/15
$${A}={hb} \\ $$$${When}\:{one}\:{variable}\:{changes} \\ $$$${Let}\:{b}\:{is}\:{changed}\:{by}\:\delta{b}\:{and}\:{corresponding}\:{change} \\ $$$${in}\:{A}\:{is}\:{denoted}\:{by}\:\:\delta{A} \\ $$$${A}={hb}\Rightarrow{A}+\delta{A}={h}\left({b}+\delta{b}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\delta{A}={h}\left({b}+\delta{b}\right)−{hb} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={hb}+{h}\delta{b}−{hb}={h}\delta{b} \\ $$$${When}\:{both}\:{variables}\:{change} \\ $$$${A}={hb}\Rightarrow{A}+\delta{A}=\left({h}+\delta{h}\right)\left({b}+\delta{b}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\delta{A}=\left({h}+\delta{h}\right)\left({b}+\delta{b}\right)−{hb} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={hb}+{h}\delta{b}+{b}\delta{h}−{hb} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={h}\delta{b}+{b}\delta{h} \\ $$
Commented by Filup last updated on 11/Dec/15
$$\mathrm{Awesome}!\:\mathrm{I}\:\mathrm{was}\:\mathrm{so}\:\mathrm{curious}\:\mathrm{as}\:\mathrm{to} \\ $$$$\mathrm{how}\:\mathrm{you}\:\mathrm{would}\:\mathrm{write}\:\mathrm{it}!\:\mathrm{Good}\:\mathrm{job}!!! \\ $$