Question Number 144777 by Nhozie last updated on 29/Jun/21 Answered by som(math1967) last updated on 29/Jun/21 $$\:\left({x}+\mathrm{3}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{3} \\ $$$${if}\:{x},{y}\in{R}\:{then}\:\left({x}+\mathrm{3}\right)^{\mathrm{2}} \geqslant\mathrm{0}\:,{y}^{\mathrm{2}} \geqslant\mathrm{0} \\ $$$$\therefore\left({x}+\mathrm{3}\right)^{\mathrm{2}}…
Question Number 144772 by phally last updated on 29/Jun/21 Answered by liberty last updated on 29/Jun/21 $$\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{x}} −\mathrm{1}\right)}{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ln}\:\left(\mathrm{1}−\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}\right)\right.}\: \\ $$$$\mathrm{consider}\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{x}} −\mathrm{1}\right)=\mathrm{x}^{\mathrm{x}} \left(\mathrm{ln}\:\mathrm{x}+\mathrm{1}\right) \\…
Question Number 13695 by Joel577 last updated on 22/May/17 $$\mathrm{Volume}\:\mathrm{of}\:\mathrm{a}\:\mathrm{bubble}\:\mathrm{is}\:\mathrm{3}\:\mathrm{times}\:\mathrm{larger} \\ $$$$\mathrm{when}\:\mathrm{it}\:\mathrm{reaches}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{from}\:\mathrm{the}\:\mathrm{bottom} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{lake}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lake}? \\ $$$$ \\ $$$$\left(\mathrm{A}\right)\:\mathrm{10}\:\mathrm{m}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{40}\:\mathrm{m} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{20}\:\mathrm{m}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{E}\right)\:\mathrm{50}\:\mathrm{m} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{30}\:\mathrm{m} \\…
Question Number 144753 by mondlihk last updated on 28/Jun/21 Answered by ajfour last updated on 28/Jun/21 $$\frac{\partial{f}}{\partial{x}}=\mathrm{3}{x}^{\mathrm{2}} −\mathrm{3}{ay}=\mathrm{0} \\ $$$$\Rightarrow\:\:{y}=\frac{{x}^{\mathrm{2}} }{{a}} \\ $$$${similarly}\:\:\frac{\partial{f}}{\partial{y}}=\mathrm{3}{y}^{\mathrm{2}} −\mathrm{3}{ax}=\mathrm{0} \\…
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Question Number 79181 by naka3546 last updated on 23/Jan/20 $$\underset{{x}\rightarrow\:\mathrm{0}^{+} } {\mathrm{lim}}\:\:\left({x}^{\mathrm{2}} \:+\:\mathrm{1}\right)^{\mathrm{ln}\:{x}} \:\:=\:\:… \\ $$ Commented by john santu last updated on 23/Jan/20 $$\:\underset{{x}\rightarrow\mathrm{0}^{+}…
Question Number 13636 by chux last updated on 21/May/17 $$\mathrm{A}\:\mathrm{steam}\:\mathrm{envine}\:\mathrm{of}\:\mathrm{efficiency}\:\mathrm{70\%} \\ $$$$\mathrm{burns}\:\mathrm{20g}\:\mathrm{of}\:\mathrm{coal}\:\mathrm{to}\:\mathrm{produce}\:\mathrm{10kJ} \\ $$$$\mathrm{of}\:\mathrm{energy}.\mathrm{If}\:\mathrm{it}\:\mathrm{burns}\:\mathrm{200g}\:\mathrm{of}\:\mathrm{coal}\: \\ $$$$\mathrm{per}\:\mathrm{second},\mathrm{calculate}\:\mathrm{its}\:\mathrm{output} \\ $$$$\mathrm{power}. \\ $$$$ \\ $$$$ \\ $$$$ \\…
Question Number 144697 by ZiYangLee last updated on 28/Jun/21 $$\mathrm{Evaluate}\:\left(\frac{\mathrm{1}+\mathrm{cos}\:\frac{\pi}{\mathrm{10}}−{i}\mathrm{sin}\:\frac{\pi}{\mathrm{10}}}{\mathrm{1}+\mathrm{cos}\:\frac{\pi}{\mathrm{10}}+{i}\mathrm{sin}\:\frac{\pi}{\mathrm{10}}}\right)^{\mathrm{15}} . \\ $$ Answered by mathmax by abdo last updated on 28/Jun/21 $$\mathrm{1}+\mathrm{cos}\left(\frac{\pi}{\mathrm{10}}\right)+\mathrm{isin}\left(\frac{\pi}{\mathrm{10}}\right)=\mathrm{2cos}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{20}}\right)+\mathrm{2isin}\left(\frac{\pi}{\mathrm{20}}\right)\mathrm{cos}\left(\frac{\pi}{\mathrm{20}}\right) \\…
Question Number 144683 by lapache last updated on 27/Jun/21 $${Determiner}\:{l}'{original}\:{de}\:{laplace} \\ $$$${F}\left({p}\right)=\frac{\mathrm{1}}{\left({p}^{\mathrm{2}} +{p}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$ Answered by Olaf_Thorendsen last updated on 27/Jun/21 $$ \\…