Question Number 8757 by FilupSmith last updated on 26/Oct/16 $$\mathrm{A}\:\mathrm{balloon}\:\mathrm{is}\:\mathrm{inflated}\:\mathrm{such}\:\mathrm{that}\:\mathrm{every} \\ $$$$\mathrm{point}\:\mathrm{expands}\:\mathrm{at}\:{a}\:\mathrm{units}/\mathrm{second}. \\ $$$$ \\ $$$$\mathrm{An}\:\mathrm{ant}\:\mathrm{runs}\:\mathrm{from}\:\mathrm{one}\:\mathrm{point}\:\boldsymbol{{A}}\:\mathrm{to}\:\mathrm{another} \\ $$$$\mathrm{point}\:\boldsymbol{{B}}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{ant}\:\mathrm{moves}\:{b}\:\mathrm{units}/\mathrm{second}, \\ $$$$\mathrm{what}\:\mathrm{will}\:\mathrm{influence}\:\mathrm{if}\:\mathrm{or}\:\mathrm{not}\:\mathrm{the}\:\mathrm{ant}\:\mathrm{will} \\ $$$$\mathrm{ever}\:\mathrm{reach}\:\mathrm{point}\:\boldsymbol{{B}}? \\ $$ Commented…
Question Number 74293 by Mr. K last updated on 21/Nov/19 Commented by Mr. K last updated on 21/Nov/19 $${ABCD}\:{id}\:{a}\:{quadrilateral},\:{cos}\theta=\frac{\sqrt{\mathrm{7}}}{\mathrm{4}}. \\ $$$${AE}=\mathrm{1},\:{BE}=\mathrm{4},\:{CE}=\mathrm{3}\:{and}\:{DE}=\mathrm{2}. \\ $$$${Find}\:{the}\:{area}\:{of}\:{the}\:{quadrilateral}. \\ $$…
Question Number 8756 by trapti rathaur@ gmail.com last updated on 25/Oct/16 $${show}\:{that}\:{every}\:{sphere}\:{through}\:{the}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{ax}+{r}^{\mathrm{2}} =\mathrm{0},{z}=\mathrm{0} \\ $$$$,{z}=\mathrm{0}\:\:\:\:\:\:\:{cuts}\:{orthogonally}\:{every}\:{sphere}\:{through}\:{the}\:{circle}\: \\ $$$${x}^{\mathrm{2}} +{z}^{\mathrm{2}} ={r}^{\mathrm{2}} ,\:{y}={o}\:. \\ $$ Terms…
Question Number 139825 by mathdave last updated on 01/May/21 Commented by mr W last updated on 01/May/21 $${rate}\:{of}\:{changing}\:{of}\:{distance}\:{is}\:{the} \\ $$$${sum}\:{of}\:{the}\:{speeds}\:{of}\:{the}\:{cars}: \\ $$$${case}\:\mathrm{1}: \\ $$$$−\left(\mathrm{50}+\mathrm{40}\right)=−\mathrm{90}\:{km}/{h}\:{if}\:{A}\:{was}\:{to} \\…
Question Number 8755 by trapti rathaur@ gmail.com last updated on 25/Oct/16 $${find}\:{the}\:{equation}\:{of}\:{the}\:{sphere}\:{which}\:{touches}\:{the}\:{plane}\: \\ $$$$\mathrm{3}{x}+\mathrm{2}{y}−{z}+\mathrm{2}=\mathrm{0}\:{at}\:{the}\:{point}\:\left(\mathrm{1},−\mathrm{2},\mathrm{1}\right)\:{and}\:{cuts}\:{orthogonally}\:{the} \\ $$$${the}\:{sphere}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{6}{y}+\mathrm{4}=\mathrm{0} \\ $$ Terms of Service Privacy…
Question Number 139824 by mathsuji last updated on 01/May/21 $${Solve}\:{in}\:\mathbb{R}\:{the}\:{following}\:{equation}: \\ $$$$\mathrm{2}\centerdot\mathrm{3}^{{x}} +\mathrm{5}\centerdot\mathrm{4}^{{x}} =\mathrm{4}\centerdot\mathrm{5}^{{x}} +\mathrm{3}\centerdot\mathrm{2}^{{x}} \\ $$ Answered by MJS_new last updated on 02/May/21 $$\mathrm{obviously}\:{x}=\mathrm{0}\vee{x}=\mathrm{1}…
Question Number 139826 by qaz last updated on 01/May/21 $$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{sin}\:\left[\left({n}−\mathrm{1}\right){x}\right]}{\mathrm{4}^{{n}+\mathrm{1}} }=? \\ $$ Answered by mnjuly1970 last updated on 01/May/21 $$\:\:\:\:\:\Omega:=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{sin}\left(\left({n}−\mathrm{1}\right){x}\right)}{\mathrm{4}^{{n}+\mathrm{1}}…
Question Number 8750 by tawakalitu last updated on 25/Oct/16 $$\mathrm{wx}\:+\:\mathrm{2z}\:=\:\mathrm{3}\:\:\:…………\:\left(\mathrm{i}\right) \\ $$$$\mathrm{3x}\:−\:\mathrm{y}\:+\:\mathrm{4z}\:=\:\mathrm{4}\:\:\:………..\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{6x}\:+\:\mathrm{2wy}\:=\:−\:\mathrm{4}\:\:\:………..\:\left(\mathrm{iii}\right) \\ $$$$ \\ $$$$\mathrm{find}\:\:\mathrm{w},\:\mathrm{x},\:\mathrm{y},\:\mathrm{z} \\ $$ Commented by Rasheed Soomro last…
Question Number 74284 by arthur.kangdani@gmail.com last updated on 21/Nov/19 Commented by mr W last updated on 21/Nov/19 $${a}_{\mathrm{2}} =\mathrm{4} \\ $$$${a}_{\mathrm{21}} =\mathrm{99} \\ $$$$\Sigma=\frac{\left(\mathrm{4}+\mathrm{99}\right)×\mathrm{20}}{\mathrm{2}}=\mathrm{1030} \\…
Question Number 74280 by ~blr237~ last updated on 21/Nov/19 $${Let}\:\:{consider}\:\alpha\::\:{I}\rightarrow\mathbb{R}^{\mathrm{2}} \:\:{a}\:{parametric}\:{curve}\:{defined}\:{as} \\ $$$$\forall\:{t}\in{I}\:\:\:\alpha\left({t}\right)=\left(\frac{{t}^{\mathrm{2}} −\mathrm{1}}{{t}^{\mathrm{3}} −\mathrm{1}}\:,\frac{\mathrm{2}{t}}{{t}^{\mathrm{3}} −\mathrm{1}}\right)\: \\ $$$${Prove}\:{that}\:{for}\:{a},{b},{c}\in{I}\:\:\: \\ $$$$\:\:\alpha\left({a}\right),\alpha\left({b}\right),\alpha\left({c}\right)\:{are}\:{on}\:{the}\:{same}\:{lign}\:{iff}\:\:{abc}={a}+{b}+{c}+\mathrm{1} \\ $$ Commented by MJS…