Question Number 132285 by bemath last updated on 13/Feb/21 $$\mathrm{Simplify}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\: \\ $$$$\frac{\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} \right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}\right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)}{\left(\mathrm{x}^{\frac{\mathrm{4}}{\mathrm{3}}} −\mathrm{x}\right)\left(\mathrm{x}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)} \\ $$$$\mathrm{with}\:\mathrm{x}\:\neq\:\mathrm{0} \\ $$$$…
Question Number 66751 by John Kaloki Musau last updated on 19/Aug/19 $${A}\:{rostum}\:{is}\:{made}\:{by}\:{cutting}\:{off} \\ $$$${the}\:{upper}\:{part}\:{of}\:{a}\:{cone}\:{along}\:{a} \\ $$$${plane}\:{parallel}\:{to}\:{the}\:{base}\:{at}\:\frac{\mathrm{2}}{\mathrm{3}}\:{up}\: \\ $$$${the}\:{height}.\:{What}\:{fraction}\:{of}\:{the}\: \\ $$$${volume}\:{of}\:{the}\:{cone}\:{does}\:{the} \\ $$$${rostum}\:{represent}? \\ $$ Answered…
Question Number 132287 by benjo_mathlover last updated on 13/Feb/21 Answered by Olaf last updated on 13/Feb/21 $$\mathrm{Let}\:{q}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{6}} {f}\left({x}\right){dx}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{6}} {f}\left({x}−\mathrm{4}\right){dx}\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{Let}\:{u}\:=\:{x}−\mathrm{4} \\ $$$$\left(\mathrm{1}\right)\::\:{q}\:=\:\mathrm{2}\int_{−\mathrm{4}}…
Question Number 66746 by John Kaloki Musau last updated on 20/Aug/19 $${A}\:{point}\:{T}\:\:{divides}\:{a}\:{line}\:{AB}\:{internally}\:{in}\:{the}\:{ratio}\:\mathrm{5}:\mathrm{2}.\:{Given}\:{that}\:{A}\:{is}\:\left(-\mathrm{4},\mathrm{10}\right)\:{and}\:{B}\:{is}\:\left(\mathrm{10},\mathrm{3}\right),\:{find}\:{the}\:{coordinates}\:{of}\:{T}. \\ $$ Answered by mr W last updated on 19/Aug/19 $${x}_{{T}} ={x}_{{A}} +\frac{\mathrm{5}}{\mathrm{7}}\left({x}_{{B}}…
Question Number 1211 by 112358 last updated on 14/Jul/15 $${Is}\:{there}\:{a}\:{solution}\:{of}\:{y}\:{in}\:{terms} \\ $$$${of}\:{x}\:{for}\:{the}\:{following}\:{D}.{E}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{dy}}{{dx}}+\frac{{c}_{\mathrm{1}} }{{y}\left({c}_{\mathrm{2}} {x}+{c}_{\mathrm{3}} \right)^{\mathrm{2}} }={c}_{\mathrm{4}} \\ $$$${Here}\:{c}_{\mathrm{1}} ,\:{c}_{\mathrm{2}} ,\:{c}_{\mathrm{3}} ,\:{c}_{\mathrm{4}} \:{are}\:{constants}.\: \\…
Question Number 1208 by 123456 last updated on 14/Jul/15 $$\mathrm{wich}\:\mathrm{statment}\:\mathrm{is}\:\mathrm{true} \\ $$$$\mathrm{1}.\mathrm{if}\:{x}\in\mathbb{Q}\:\mathrm{and}\:{y}\in\mathbb{Z},\:\mathrm{then}\:{xy}\in\mathbb{Q}/\mathbb{Z} \\ $$$$\mathrm{2}.\mathrm{if}\:{x}\in\mathbb{Q}/\mathbb{Z}\:\mathrm{and}\:{y}\in\mathbb{Z},\:\mathrm{then}\:{xy}\in\mathbb{Q}/\mathbb{Z} \\ $$$$\mathrm{3}.\mathrm{if}\:{x}\in\mathbb{R},\:\mathrm{then}\:\exists{y}\in\mathbb{R}\:\mathrm{such}\:\mathrm{that}\:{xy}=\mathrm{5} \\ $$$$\mathrm{4}.\mathrm{if}\:{x}\in\mathbb{N}/\left\{\mathrm{0}\right\},\:\mathrm{then}\:\exists{y}\in\mathbb{Q}\:\mathrm{such}\:\mathrm{that}\:\sqrt{\frac{{y}\left({x}+\mathrm{2}\right)}{{x}\left({x}+\mathrm{1}\right)}}\in\mathbb{Q} \\ $$ Answered by prakash jain last…
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Question Number 66743 by pete last updated on 19/Aug/19 $$\mathrm{Each}\:\mathrm{month}\:\mathrm{a}\:\mathrm{store}\:\mathrm{owner}\:\mathrm{can}\:\mathrm{spend}\:\mathrm{at} \\ $$$$\mathrm{most}\:\$\mathrm{100},\mathrm{000}\:\mathrm{on}\:\mathrm{PC}'\mathrm{s}\:\mathrm{and}\:\mathrm{laptops}.\:\mathrm{A} \\ $$$$\mathrm{PC}\:\mathrm{costs}\:\mathrm{the}\:\mathrm{store}\:\mathrm{owner}\:\$\mathrm{1000}\:\mathrm{and}\:\mathrm{a} \\ $$$$\mathrm{laptop}\:\mathrm{costs}\:\mathrm{him}\:\$\mathrm{1500}.\:\mathrm{Each}\:\mathrm{PC}\:\mathrm{is}\:\mathrm{sold} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\$\mathrm{400}\:\mathrm{while}\:\mathrm{a}\:\mathrm{laptop}\:\mathrm{is}\:\mathrm{sold} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\$\mathrm{700}.\:\mathrm{The}\:\mathrm{store}\:\mathrm{owner}\:\mathrm{estimates} \\ $$$$\mathrm{that}\:\mathrm{at}\:\mathrm{least}\:\mathrm{15}\:\mathrm{PC}'\mathrm{s}\:\mathrm{but}\:\mathrm{no}\:\mathrm{more}\:\mathrm{than} \\ $$$$\mathrm{80}\:\mathrm{are}\:\mathrm{sold}\:\mathrm{each}\:\mathrm{month}.\:\mathrm{He}\:\mathrm{also}\:\mathrm{estimates} \\…
Question Number 1206 by gs8763330@gmail.com last updated on 14/Jul/15 $${prove}\:{that}\:\:\mathrm{2sin}^{−\mathrm{1}} \mathrm{3}/\mathrm{4}=\mathrm{tan}^{−\mathrm{1}} \mathrm{24}/\mathrm{7} \\ $$ Commented by 123456 last updated on 14/Jul/15 $$\mathrm{sin}\:\alpha={a}\overset{{b}.{c}} {\Leftrightarrow}\alpha=\mathrm{sin}^{−\mathrm{1}} {a} \\…
Question Number 66740 by behi83417@gmail.com last updated on 19/Aug/19 $$\underset{\:\:\mathrm{0}} {\overset{\:\:\:\:\:\:\:\mathrm{1}} {\int}}\sqrt{\mathrm{1}−\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:}\:\:\boldsymbol{\mathrm{dx}}=? \\ $$ Commented by mathmax by abdo last updated on 21/Aug/19…