Question Number 4543 by FilupSmith last updated on 06/Feb/16 $$\mathrm{Triangle}\:{ABC}\:\mathrm{has}\:\mathrm{midpoints} \\ $$$${D},\:{E}\:\mathrm{and}\:{F}. \\ $$$$ \\ $$$$\mathrm{By}\:\mathrm{connecting}\:\mathrm{each}\:\mathrm{verticie}\:\mathrm{with}\:\mathrm{the} \\ $$$$\mathrm{opposite}\:\mathrm{midpoint},\:\mathrm{we}\:\mathrm{create}\:\mathrm{a}\:\mathrm{cress}−\mathrm{section} \\ $$$$\mathrm{called}\:{G}. \\ $$$$ \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{all}\:\mathrm{three}\:\mathrm{lines}\:\mathrm{cross}\:\mathrm{at}\: \\…
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Question Number 4540 by Yozzii last updated on 06/Feb/16 $${Let}\:{us}\:{define}\:{the}\:{positive}\:{number}\:{n}\:{with}\:{four} \\ $$$${digits}\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}},\boldsymbol{\mathrm{c}}\:{and}\:\boldsymbol{\mathrm{d}}\:{such}\:{that}\:{n}=\boldsymbol{\mathrm{abcd}} \\ $$$${with}\:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}},\boldsymbol{\mathrm{c}},\boldsymbol{\mathrm{d}}\in\mathbb{Z},\:\mathrm{1}\leqslant\boldsymbol{\mathrm{a}}\leqslant\mathrm{9},\:\mathrm{0}\leqslant\boldsymbol{\mathrm{b}}\leqslant\mathrm{9}, \\ $$$$\mathrm{0}\leqslant\boldsymbol{\mathrm{c}}\leqslant\mathrm{9}\:{and}\:\mathrm{0}\leqslant\boldsymbol{\mathrm{d}}\leqslant\mathrm{9}.\:{Let}\:{us}\:{then}\:{say} \\ $$$${that}\:{a}\:{cool}\:{number}\:{is}\:{a}\:{four}\:{digit}\:{number}, \\ $$$${say}\:{n},\:{such}\:{that}\:{the}\:{two}\:{digit}\:{numbers}\:{written}\:{as} \\ $$$$\boldsymbol{\mathrm{ab}}\:{and}\:\boldsymbol{\mathrm{cd}}\:{are}\:{given}\:{by}\:\boldsymbol{\mathrm{ab}}={r}×{s}\:{and} \\ $$$$\boldsymbol{\mathrm{cd}}=\left({r}−\mathrm{1}\right)×\left({s}+\mathrm{1}\right)\:{for}\:{some}\:{non}−{negative}\:{integers} \\…
Question Number 135609 by physicstutes last updated on 14/Mar/21 $$\mathrm{express}\:\mathrm{the}\:\mathrm{number}\:\frac{\mathrm{3}}{\mathrm{13}}\:\mathrm{as}\:\mathrm{a}\:\mathrm{decimal}\:\mathrm{ane}\:\mathrm{state}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{300}^{\mathrm{th}} \:\mathrm{value} \\ $$ Answered by Olaf last updated on 14/Mar/21 $$\frac{\mathrm{3}}{\mathrm{13}}\:=\:\mathrm{0},\overline {\mathrm{230769}} \\…
Question Number 70074 by cat2315 last updated on 30/Sep/19 $$\int_{\mathrm{1}} ^{\mathrm{2}} \left[\mathrm{3}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right]{dt}= \\ $$ Commented by mathmax by abdo last updated on 30/Sep/19 $${if}\:\left[..\right]{means}\:{the}\:{integr}\:{part}\:\:{we}\:{get}…
Question Number 4539 by love math last updated on 05/Feb/16 $${Two}\:{tractor},\:{working}\:{together},\: \\ $$$${have}\:{plowed}\:{the}\:{field}\:{for}\:\mathrm{48}\:{hours}. \\ $$$${If}\:{half}\:{of}\:{the}\:{field}\:{is}\:{plowed}\:{by}\: \\ $$$${one}\:{of}\:{them}\:{and}\:{then}\:{the}\:{remaining} \\ $$$${half}\:{to}\:{the}\:{other},\:{the}\:{work}\:{would} \\ $$$${be}\:{carried}\:{out}\:\mathrm{100}\:{hours}.\:{How}\:{many} \\ $$$${hours}\:{to}\:{plow}\:{the}\:{field}\:{if}\:{every}\:{tractor} \\ $$$${working}\:{separately}?…
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Question Number 70075 by ajfour last updated on 30/Sep/19 Commented by behi83417@gmail.com last updated on 30/Sep/19 Commented by behi83417@gmail.com last updated on 30/Sep/19 $$\measuredangle\mathrm{ACB}=\frac{\mathrm{540}}{\mathrm{5}}=\mathrm{108}^{\bullet} \\…
Question Number 135610 by mnjuly1970 last updated on 14/Mar/21 $$\:\:\:\:\:\:\:\:\:\:….\:\mathscr{A}{dvanced}\:\:……\:\:\mathscr{C}{alculus}…. \\ $$$$\:\:\:\:\:\:\:\:\:{prove}\:{that}\:: \\ $$$$\:\:\:\begin{array}{|c|c|}{{i}\:::\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\right)\:={cos}\left(\frac{\pi{x}}{\mathrm{2}}\right)\:\:\:\:\checkmark\:\:}\\{{ii}\:::\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\right)=\:{cosh}\left(\frac{\pi{x}}{\mathrm{2}}\right)\:\checkmark\checkmark}\\\hline\end{array}\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…………. \\…
Question Number 4535 by FilupSmith last updated on 05/Feb/16 $$\mathrm{Lets}\:\mathrm{say}\:\mathrm{we}\:\mathrm{have}\:\mathrm{three}\:\mathrm{points}: \\ $$$${A}\left(\mathrm{0},\:\mathrm{0}\right) \\ $$$${B}\left({x},\:{y}\right) \\ $$$${C}\left(\delta{x},\:\delta{y}\right) \\ $$$$ \\ $$$$\mathrm{Assuming}\:\mathrm{that}\:\mathrm{both}\:{B}\:\mathrm{and}\:{C}\:\mathrm{are}\:\mathrm{point} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{fuction}\:{y}={f}\left({x}\right),\:\mathrm{we}\:\mathrm{can}\:\mathrm{calculate} \\ $$$$\mathrm{the}\:\mathrm{area}\:\mathrm{under}\:\mathrm{the}\:\mathrm{point}\:\mathrm{where}\:\mathrm{it}\:\mathrm{makes} \\…