Question Number 132440 by liberty last updated on 14/Feb/21 $$\mathrm{A}\:\mathrm{vessel}\:\mathrm{containing}\:\mathrm{water}\:\mathrm{has}\:\mathrm{the}\: \\ $$$$\mathrm{shape}\:\mathrm{of}\:\mathrm{and}\:\mathrm{inverted}\:\mathrm{right}\:\mathrm{circular} \\ $$$$\mathrm{cone}\:\mathrm{with}\:\mathrm{base}\:\mathrm{radius}\:\mathrm{2m}\:\mathrm{and}\:\mathrm{height}\:\mathrm{5m} \\ $$$$\mathrm{The}\:\mathrm{water}\:\mathrm{flows}\:\mathrm{from}\:\mathrm{the}\:\mathrm{apex} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{at}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{rate}\: \\ $$$$\mathrm{of}\:\mathrm{0}.\mathrm{2}\:\mathrm{m}^{\mathrm{3}} /\mathrm{min}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{at} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{water}\:\mathrm{level}\:\mathrm{is}\:\mathrm{dropping} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{water}\:\mathrm{is}…
Question Number 132407 by bramlexs22 last updated on 14/Feb/21 $$\mathrm{on}\:\mathrm{circle}\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{25}\:,\:\mathrm{find}\:\mathrm{the}\:\mathrm{point} \\ $$$$\mathrm{closest}\:\mathrm{to}\:\left(\mathrm{1},\mathrm{1}\right). \\ $$ Answered by EDWIN88 last updated on 14/Feb/21 $$\mathrm{let}\:\mathrm{P}\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circle}\: \\…
Question Number 66866 by rajesh4661kumar@gmail.com last updated on 20/Aug/19 Commented by mathmax by abdo last updated on 20/Aug/19 $${first}\:\:{x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }>\mathrm{0}\:\:{for}\:{all}\:{x}\:{so}\:{y}={ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)={argsh}\left({x}\right)\:\Rightarrow \\ $$$${y}^{'} \left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:=\left(\mathrm{1}+{x}^{\mathrm{2}}…
Question Number 132225 by bemath last updated on 12/Feb/21 $$ \\ $$$$\rightarrow\mathrm{A}\:\mathrm{lighthouse}\:\mathrm{L}\:\mathrm{is}\:\mathrm{located}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{small}\:\mathrm{island}\:\mathrm{2}\:\mathrm{km}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{nearest}\:\mathrm{point}\:\mathrm{A}\:\mathrm{on}\:\mathrm{a}\:\mathrm{long}\:\mathrm{the} \\ $$$$\mathrm{straigh}\:\mathrm{shoreline}\:.\:\mathrm{If}\:\mathrm{the} \\ $$$$\mathrm{lighthouse}\:\mathrm{lamp}\:\mathrm{rotates}\:\mathrm{at}\:\mathrm{3} \\ $$$$\mathrm{revolutions}\:\mathrm{per}\:\mathrm{minute}.\:\mathrm{how}\:\mathrm{fast}\: \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{illuminated}\:\mathrm{spot}\:\mathrm{P}\:\mathrm{on}\:\mathrm{the} \\…
Question Number 132211 by benjo_mathlover last updated on 12/Feb/21 $$ \\ $$$$\mathrm{how}\:\mathrm{fast}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\: \\ $$$$\mathrm{rectangle}\:\mathrm{changing}\:\mathrm{if}\:\mathrm{one}\:\mathrm{side}\:\mathrm{is}\:\mathrm{10} \\ $$$$\:\mathrm{cm}\:\mathrm{long}\:\mathrm{and}\:\mathrm{increasing}\:\mathrm{at}\:\mathrm{a} \\ $$$$\mathrm{rate}\:\mathrm{of}\:\mathrm{2}\:\mathrm{cm}/\mathrm{s}\:\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{side}\:\mathrm{is}\: \\ $$$$\mathrm{8}\:\mathrm{cm}\:\mathrm{long}\:\mathrm{and}\:\mathrm{is}\:\mathrm{decreasing}\:\mathrm{at}\: \\ $$$$\mathrm{a}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{3}\:\mathrm{cm}/\mathrm{s} \\ $$ Answered…
Question Number 1130 by 123456 last updated on 22/Jun/15 $${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({xy}\right)+{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right)+{f}\left({x}\right)+{f}\left({y}\right) \\ $$$${f}\left(−\mathrm{1}\right)=? \\ $$$${f}\left(\mathrm{0}\right)=? \\ $$$${f}\left(+\mathrm{1}\right)=? \\ $$ Answered by prakash jain last…
Question Number 132104 by mnjuly1970 last updated on 11/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{calculus}\:\left(\mathrm{1}\right)… \\ $$$$\:\:{if}\:\:{y}=\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{{sin}\left({x}\right)+{cos}\left({x}\right)}}\:\:{then}\:: \\ $$$$\:\:\:\:\mathrm{3}{y}''−\mathrm{12}\left({y}'\right)^{\mathrm{2}} −{y}^{\mathrm{2}} =\:??? \\ $$$$\:\:\:\:\:……… \\ $$ Answered by mnjuly1970 last updated…
Question Number 132076 by liberty last updated on 11/Feb/21 $$\mathrm{Given}\:\mathrm{the}\:\mathrm{function}\:\mathrm{f}\left({x}\right)=\int_{\mathrm{0}} ^{\:{x}} \left(\mathrm{1}+{t}^{\mathrm{3}} \right)^{−\mathrm{1}/\mathrm{2}} {dt}. \\ $$$$\mathrm{If}\:\mathrm{h}\left({x}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{inverse}\:\mathrm{of}\:\mathrm{f}\left({x}\right)\:\mathrm{and}\:\mathrm{h}'\left({x}\right) \\ $$$$\mathrm{is}\:\mathrm{derivative}\:\mathrm{of}\:\mathrm{h}\left({x}\right).\:\mathrm{Find}\:\mathrm{the}\: \\ $$$$\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{h}''\left({x}\right)}{\left(\mathrm{h}\left(\mathrm{x}\right)\right)^{\mathrm{2}} }\:. \\ $$$$ \\ $$…
Question Number 66498 by miracle wokama last updated on 16/Aug/19 Commented by MJS last updated on 16/Aug/19 $$\mathrm{too}\:\mathrm{complicated} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{see}\:\mathrm{that}\:{f}'\left(\mathrm{0}\right)=\mathrm{1}\:\mathrm{and}\:{f}'\left(\mathrm{1}\right)=\mathrm{100} \\ $$ Commented by mathmax…
Question Number 923 by 123456 last updated on 25/Apr/15 $${f}\left({x},{y},{z}\right)={x}+{y}+{z} \\ $$$$\boldsymbol{{g}}\left({x},{y},{z}\right)=\left({x},{y},{z}\right) \\ $$$$\boldsymbol{{h}}\left({x},{y},{z}\right)=\boldsymbol{{g}}\left({x},{y},{z}\right)−\bigtriangledown{f}\left({x},{y},{z}\right)=??? \\ $$$$\bigtriangledown\centerdot\boldsymbol{{h}}\left({x},{y},{z}\right)=? \\ $$$$\bigtriangledown×\boldsymbol{{h}}\left({x},{y},{z}\right)=??? \\ $$ Answered by 2closedStringsMeet last updated…