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Question Number 121985 Answers: 0 Comments: 1

Σ_(n=1) ^∞ (1/(n (((4n)),((2n)) )))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{n}\begin{pmatrix}{\mathrm{4n}}\\{\mathrm{2n}}\end{pmatrix}} \\ $$

Question Number 121802 Answers: 1 Comments: 2

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Question Number 121539 Answers: 3 Comments: 1

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Question Number 120244 Answers: 3 Comments: 2

how to justify that sin (x−((7π)/2) )= cos x

$${how}\:{to}\:{justify}\:\:{that}\:\mathrm{sin}\:\left({x}−\frac{\mathrm{7}\pi}{\mathrm{2}}\:\right)=\:\mathrm{cos}\:{x} \\ $$

Question Number 120215 Answers: 1 Comments: 0

Prove that (((((a+b)(b+c)(c+a))/8) ))^(1/3) ≥ (√((ab+bc+ca)/3)) for a,b,c > 0

$${Prove}\:{that}\:\sqrt[{\mathrm{3}}]{\frac{\left({a}+{b}\right)\left({b}+{c}\right)\left({c}+{a}\right)}{\mathrm{8}}\:}\:\geqslant\:\sqrt{\frac{{ab}+{bc}+{ca}}{\mathrm{3}}} \\ $$$${for}\:{a},{b},{c}\:>\:\mathrm{0} \\ $$

Question Number 120133 Answers: 0 Comments: 5

Question Number 119812 Answers: 2 Comments: 0

Question Number 119503 Answers: 2 Comments: 0

solve ⌊ (√x) ⌋ = ⌊ (x)^(1/(3 )) ⌋

$${solve}\:\lfloor\:\sqrt{{x}}\:\rfloor\:=\:\lfloor\:\sqrt[{\mathrm{3}\:}]{{x}}\:\rfloor\: \\ $$

Question Number 119187 Answers: 0 Comments: 3

Question Number 118811 Answers: 1 Comments: 4

Question Number 118545 Answers: 1 Comments: 1

Question Number 118251 Answers: 1 Comments: 0

Question Number 118138 Answers: 2 Comments: 0

Question Number 118104 Answers: 0 Comments: 1

1.) A right circular cone is circumscribed about a sphere of radius(r). If d is the distance from the center of the sphere to the vertex of the cone, show that the volume of the cone,V=((𝛑r^2 (r+d)^2 )/(3(d−r))). 2.) Find the vertical angle of the cone when it′s volume is minimum.

$$\left.\mathrm{1}.\right) \\ $$$$\mathrm{A}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{circumscribed} \\ $$$$\mathrm{about}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{radius}\left(\boldsymbol{\mathrm{r}}\right).\:\:\mathrm{If}\:\boldsymbol{\mathrm{d}}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\boldsymbol{\mathrm{V}}=\frac{\boldsymbol{\pi\mathrm{r}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{r}}+\boldsymbol{\mathrm{d}}\right)^{\mathrm{2}} }{\mathrm{3}\left(\boldsymbol{\mathrm{d}}−\boldsymbol{\mathrm{r}}\right)}. \\ $$$$\left.\mathrm{2}.\right) \\ $$$$\boldsymbol{\mathrm{F}}\mathrm{ind}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{when} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{volume}\:\mathrm{is}\:\mathrm{minimum}. \\ $$

Question Number 118069 Answers: 1 Comments: 0

1.)i) A right circular cone is circumscribed about a sphere of radius(r). If d is the distance from the center of the sphere to the vertex of the cone, show that the volume of the cone,V=((𝛑r^2 (r+d)^2 )/(3(d−r))). ii) Find the vertical angle of the cone when it′s volume is minimum.

$$\left.\mathrm{1}\left..\right)\mathrm{i}\right)\:\mathrm{A}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{circumscribed} \\ $$$$\mathrm{about}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{radius}\left(\boldsymbol{\mathrm{r}}\right).\:\:\mathrm{If}\:\boldsymbol{\mathrm{d}}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\boldsymbol{\mathrm{V}}=\frac{\boldsymbol{\pi\mathrm{r}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{r}}+\boldsymbol{\mathrm{d}}\right)^{\mathrm{2}} }{\mathrm{3}\left(\boldsymbol{\mathrm{d}}−\boldsymbol{\mathrm{r}}\right)}. \\ $$$$\left.\mathrm{ii}\right) \\ $$$$\boldsymbol{\mathrm{F}}\mathrm{ind}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{when} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{volume}\:\mathrm{is}\:\mathrm{minimum}. \\ $$

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