Question Number 137327 by mr W last updated on 01/Apr/21 $${in}\:{triangle}\:\Delta{ABC}:\:{BC}=\mathrm{1},\:\angle{B}=\mathrm{2}\angle{A}. \\ $$$${find}\:{the}\:{maximum}\:{area}\:{of}\:\Delta{ABC}. \\ $$ Answered by EDWIN88 last updated on 01/Apr/21 $$\angle\mathrm{A}+\angle\mathrm{B}+\angle\mathrm{C}\:=\pi\:;\:\mathrm{3}\angle\mathrm{A}+\angle\mathrm{C}=\pi \\ $$$$\mathrm{let}\:\angle\mathrm{A}\:=\alpha\:;\:\angle\mathrm{B}=\mathrm{2}\alpha\:;\:\angle\mathrm{C}=\pi−\mathrm{3}\alpha…
Question Number 137321 by mathocean1 last updated on 01/Apr/21 $${H}\left({x}\right)=\frac{\mathrm{8}}{{x}−\mathrm{2}}\:{and}\:{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{6}}{\mathrm{2}{x}−\mathrm{4}} \\ $$$$\left.\mathrm{1}\right){Calculate}\:{the}\:{surface}\:{V}_{\boldsymbol{{n}}} \:{of} \\ $$$${area}\:{limited}\:{by}\:{the}\:{the}\:{line} \\ $$$${x}=\mathrm{6};\:{x}=\mathrm{6}+\boldsymbol{{n}}\:\left({n}\in\mathbb{N}^{\ast} \right)\:{and}\:{the} \\ $$$${curve}\:{of}\:{H}\left({x}\right)\:{and}\:{f}\left({x}\right)\:{in}\:{function} \\ $$$${of}\:\boldsymbol{{n}}. \\ $$$$\left.\mathrm{2}\right)\:{Knowing}\:{that}\:\mathrm{1}^{\mathrm{2}}…
Question Number 6251 by net last updated on 20/Jun/16 $$\mathrm{3}{x}−\mathrm{2}{y}−\mathrm{4}{z}=\mathrm{2} \\ $$$$\mathrm{3}{y}−\mathrm{4}{z}=−\mathrm{2} \\ $$$$\mathrm{2}{y}+\mathrm{6}{z}=−\mathrm{1} \\ $$ Answered by Rasheed Soomro last updated on 20/Jun/16 $$\mathrm{3}{x}−\mathrm{2}{y}−\mathrm{4}{z}=\mathrm{2}………\left({i}\right)…
Question Number 6248 by FilupSmith last updated on 20/Jun/16 $$\int_{−\infty} ^{\:\infty} {e}^{−{u}} {u}^{{n}} {du}\:=\:??? \\ $$ Commented by 123456 last updated on 20/Jun/16 $$\mathrm{i}\:\mathrm{think}\:\mathrm{that}\:\mathrm{it}\:\mathrm{diverges} \\…
Question Number 137317 by physicstutes last updated on 01/Apr/21 Commented by physicstutes last updated on 01/Apr/21 $$\mathrm{The}\:\mathrm{figure}\:\mathrm{above}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{composite}\:\mathrm{bar} \\ $$$$\mathrm{made}\:\mathrm{of}\:\mathrm{two}\:\mathrm{materials}\:{X}\:\mathrm{and}\:{Y}.\:\mathrm{The}\:\mathrm{end} \\ $$$$\mathrm{of}\:{X}\:\mathrm{is}\:\mathrm{maintained}\:\mathrm{at}\:\mathrm{100}°\mathrm{C}\:\mathrm{and}\:\mathrm{it}\:\mathrm{has} \\ $$$$\mathrm{length}\:\mathrm{of}\:\mathrm{25}\:\mathrm{cm}\:\mathrm{while}\:\mathrm{that}\:\mathrm{of}\:{Y}\:\mathrm{is}\:\mathrm{maintained} \\ $$$$\mathrm{at}\:\mathrm{0}°\mathrm{C}\:\mathrm{with}\:\mathrm{lenght}\:\mathrm{75}\:\mathrm{cm}.\:\mathrm{Find}\:\mathrm{the}…
Question Number 6247 by Yozzii last updated on 20/Jun/16 $${Show}\:{that},\:\forall{x}\in\mathbb{R},\:{the}\:{sequence}\:\left\{{f}\left({n}\right)\right\}_{\mathrm{0}} ^{\infty} \\ $$$${defined}\:{by}\:{f}\left(\mathrm{0}\right)={cosx},\:{f}\left(\mathrm{1}\right)={sin}\left({cosx}\right)\:{and} \\ $$$${f}\left({n}\right)=\begin{cases}{{sin}\left({cos}\left({f}\left({n}−\mathrm{2}\right)\right)\right)\:\:{if}\:{n}\geqslant\mathrm{3}\:{is}\:{odd}}\\{{cos}\left({sin}\left({f}\left({n}−\mathrm{2}\right)\right)\right)\:\:{if}\:{n}\geqslant\mathrm{2}\:{is}\:{even},}\end{cases} \\ $$$${converges}\:{to}\:{a}\:{limit}\:{l}\in\left(\mathrm{0}.\mathrm{6},\mathrm{0}.\mathrm{7}\right). \\ $$$${If}\:{you}\:{can},\:{determine}\:{the}\:{exact}\:{value} \\ $$$${of}\:{l}. \\ $$ Commented by…
Question Number 137316 by BHOOPENDRA last updated on 01/Apr/21 $${An}\:{alternating}\:{current}\:{after}\:{passing}\: \\ $$$$\:{through}\:{rectifire}\:{has}\:{the} \\ $$$${form}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{i}={I}_{\mathrm{0}} {sinx}\:\:\:\:\:\:{for}\:\mathrm{0}\leqslant{x}\leqslant\pi \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:{for}\:\pi\leqslant{x}\leqslant\mathrm{2}\pi \\ $$$${where}\:{I}_{\mathrm{0}} \:{is}\:{the}\:{maximum}\:{current}\: \\ $$$${and}\:{period}\:{is}\:\mathrm{2}\pi.{express}\:{i}\:{is}\:{a}\: \\…
Question Number 6246 by FilupSmith last updated on 20/Jun/16 $$\mathrm{Need}\:\mathrm{help}\:\mathrm{solving}\:\int_{\mathrm{0}} ^{\:\infty} {x}^{−{x}} {dx} \\ $$$$\mathrm{My}\:\mathrm{current}\:\mathrm{working}: \\ $$$${x}^{−{x}} ={e}^{−{x}\mathrm{ln}{x}} \\ $$$${e}^{−{x}\mathrm{ln}{x}} =\mathrm{1}−{x}\mathrm{ln}\left({x}\right)+\frac{{x}^{\mathrm{2}} \mathrm{ln}\left({x}\right)^{\mathrm{2}} }{\mathrm{2}!}−\frac{{x}^{\mathrm{3}} \mathrm{ln}\left({x}\right)^{\mathrm{3}} }{\mathrm{3}!}+……
Question Number 137315 by liberty last updated on 01/Apr/21 $$\int_{\mathrm{0}} ^{\:\pi/\mathrm{4}} \:\sqrt{\mathrm{cos}\:^{\mathrm{3}} \left(\mathrm{2x}\right)}\:\mathrm{dx}\:? \\ $$ Answered by rs4089 last updated on 01/Apr/21 $$\int_{\mathrm{0}} ^{\:\pi/\mathrm{4}} \sqrt{{cos}^{\mathrm{3}}…
Question Number 71776 by TawaTawa last updated on 19/Oct/19 Answered by tw000001 last updated on 22/Oct/19 $$\mathrm{Use}\:\mathrm{Harmonic}\:\mathrm{series}\:\mathrm{to}\:\mathrm{solve}. \\ $$$${A}=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}+…+\frac{\mathrm{1}}{\mathrm{2015}}−\frac{\mathrm{1}}{\mathrm{2016}} \\ $$$$=\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+…+\frac{\mathrm{1}}{\mathrm{2016}}\right)−\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{6}}+…+\frac{\mathrm{1}}{\mathrm{2016}}\right) \\ $$$$=\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+…+\frac{\mathrm{1}}{\mathrm{2016}}\right)−\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+…+\frac{\mathrm{1}}{\mathrm{1008}}\right) \\ $$$$={H}_{\mathrm{2016}}…