Question and Answers Forum

A regular pyramid has for its base polygon of n sides, and each slant face consist of an isosceles triangle of vertical angle 2α. If the slant faces are each inclined at angle β to the base , and at an angle 2γ to one another show that cosβ = tan α cot(π/n) , and sinγ = sec α cos(π/n)

$$\mathrm{A}\:\mathrm{regular}\:\mathrm{pyramid}\:\mathrm{has}\:\mathrm{for}\:\mathrm{its}\:\mathrm{base}\:\mathrm{polygon} \\ $$$$\mathrm{of}\:{n}\:\mathrm{sides},\:\mathrm{and}\:\mathrm{each}\:\mathrm{slant}\:\mathrm{face}\:\mathrm{consist}\:\mathrm{of}\:\mathrm{an}\: \\ $$$$\mathrm{isosceles}\:\mathrm{triangle}\:\mathrm{of}\:\mathrm{vertical}\:\mathrm{angle}\:\mathrm{2}\alpha.\:\mathrm{If}\:\mathrm{the} \\ $$$$\mathrm{slant}\:\mathrm{faces}\:\mathrm{are}\:\mathrm{each}\:\mathrm{inclined}\:\mathrm{at}\:\mathrm{angle}\:\beta\:\mathrm{to}\: \\ $$$$\mathrm{the}\:\mathrm{base}\:,\:\mathrm{and}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{2}\gamma\:\mathrm{to}\:\mathrm{one}\:\mathrm{another} \\ $$$$\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{cos}\beta\:=\:\mathrm{tan}\:\alpha\:\mathrm{cot}\frac{\pi}{\mathrm{n}}\:,\:\mathrm{and}\:\mathrm{sin}\gamma\:=\:\mathrm{sec}\:\alpha\:\mathrm{cos}\frac{\pi}{\mathrm{n}} \\ $$

Question Number 37864 Answers: 2 Comments: 0

prove that cos (π/(15)) cos ((2π)/(15)) cos ((3π)/(15)) cos ((4π)/(15)) cos ((5π)/(15)) cos ((6π)/(15)) cos ((7π)/(15)) = (1/2^7 )

$$\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{cos}\:\frac{\pi}{\mathrm{15}}\:\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{15}}\:\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{15}}\:\mathrm{cos}\:\frac{\mathrm{4}\pi}{\mathrm{15}}\:\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{15}}\:\mathrm{cos}\:\frac{\mathrm{6}\pi}{\mathrm{15}}\:\mathrm{cos}\:\frac{\mathrm{7}\pi}{\mathrm{15}}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{7}} } \\ $$

Question Number 37862 Answers: 1 Comments: 7

Question Number 37861 Answers: 0 Comments: 1

which is the chain rule? A. (dy/dx) = (dy/dx) × 1 B. (dy/dx) = (du/dx) × (dy/dx) C. (dy/dx) = (dy/du) × (du/dx) D. (dy/dx) = (dy/du) × (dy/dx)

$$\:\:\:{which}\:{is}\:{the}\:{chain}\:{rule}? \\ $$$${A}.\:\frac{{dy}}{{dx}}\:=\:\frac{{dy}}{{dx}}\:×\:\mathrm{1} \\ $$$${B}.\:\frac{{dy}}{{dx}}\:=\:\frac{{du}}{{dx}}\:×\:\frac{{dy}}{{dx}} \\ $$$${C}.\:\frac{{dy}}{{dx}}\:=\:\frac{{dy}}{{du}}\:×\:\frac{{du}}{{dx}} \\ $$$${D}.\:\frac{{dy}}{{dx}}\:=\:\frac{{dy}}{{du}}\:×\:\frac{{dy}}{{dx}} \\ $$

Question Number 37860 Answers: 1 Comments: 0

The distance moved by a particle in t seconds is given by s= t^3 + 3t + 1 where s is in metres.Find the velocity and Asseleration after 3 seconds. Show all steps with short statements on how the answers are gotten.

$${The}\:{distance}\:{moved}\:{by}\:{a}\:{particle} \\ $$$${in}\:{t}\:{seconds}\:{is}\:{given}\:{by} \\ $$$${s}=\:{t}^{\mathrm{3}} \:+\:\mathrm{3}{t}\:+\:\mathrm{1}\:{where}\:{s}\:{is}\:{in}\: \\ $$$${metres}.{Find}\:{the}\:{velocity}\:{and}\: \\ $$$${Asseleration}\:{after}\:\mathrm{3}\:{seconds}. \\ $$$$ \\ $$$${Show}\:{all}\:{steps}\:{with}\:{short}\:{statements} \\ $$$${on}\:{how}\:{the}\:{answers}\:{are}\:{gotten}. \\ $$

Question Number 37859 Answers: 0 Comments: 1

Σ_(k=1) ^∞ ((Π_(i=0) ^(k−1) (n−i))/(k!)) is this really mean something

$$\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\underset{{i}=\mathrm{0}} {\overset{{k}−\mathrm{1}} {\prod}}\left({n}−{i}\right)}{{k}!} \\ $$$$\mathrm{is}\:\mathrm{this}\:\mathrm{really}\:\mathrm{mean}\:\mathrm{something} \\ $$

Question Number 37845 Answers: 0 Comments: 0

Question Number 37840 Answers: 1 Comments: 0

f(θ,φ)=((cos φ[cos θ tan (((θ+φ)/2))−sin θ]^2 )/(cos φtan (((θ+φ)/2))+sin φ)) φ ∈ (0,(π/2)) , θ ∈ (−(π/2), (π/2)); find maximum f(θ,φ).

$${f}\left(\theta,\phi\right)=\frac{\mathrm{cos}\:\phi\left[\mathrm{cos}\:\theta\:\mathrm{tan}\:\left(\frac{\theta+\phi}{\mathrm{2}}\right)−\mathrm{sin}\:\theta\right]^{\mathrm{2}} }{\mathrm{cos}\:\phi\mathrm{tan}\:\left(\frac{\theta+\phi}{\mathrm{2}}\right)+\mathrm{sin}\:\phi} \\ $$$$\:\phi\:\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right)\:,\:\theta\:\in\:\left(−\frac{\pi}{\mathrm{2}},\:\frac{\pi}{\mathrm{2}}\right); \\ $$$${find}\:{maximum}\:{f}\left(\theta,\phi\right). \\ $$

Question Number 37838 Answers: 0 Comments: 5

let f(x)= e^(−2x) arctan(x^2 ) 1)calculate f^((n)) (x) 2) find f^((n)) (0) 3) developp f at integr serie

$${let}\:{f}\left({x}\right)=\:{e}^{−\mathrm{2}{x}} \:{arctan}\left({x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 37820 Answers: 1 Comments: 2

let f(x)= (x^2 /(1+x^4 )) 1) calculate f^((n)) (x) 2) find f^((n)) (0) 3) developp f at integr serie 4) calculate ∫_0 ^1 f(x)dx.

$${let}\:{f}\left({x}\right)=\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}. \\ $$

Question Number 37819 Answers: 0 Comments: 0

study the convergence of u_(n+1) =u_n + ln(1+e^(−u_n ) ) with u_0 =0

$${study}\:{the}\:{convergence}\:\:{of} \\ $$$${u}_{{n}+\mathrm{1}} ={u}_{{n}} \:+\:{ln}\left(\mathrm{1}+{e}^{−{u}_{{n}} } \right)\:\:{with}\:{u}_{\mathrm{0}} =\mathrm{0} \\ $$

Question Number 37818 Answers: 0 Comments: 1

let 0<u_0 <1 and u_(n+1) =(√((1+u_n )/2)) study the convergence of u_n

$${let}\:\mathrm{0}<{u}_{\mathrm{0}} <\mathrm{1}\:\:{and}\:{u}_{{n}+\mathrm{1}} =\sqrt{\frac{\mathrm{1}+{u}_{{n}} }{\mathrm{2}}} \\ $$$${study}\:{the}\:{convergence}\:{of}\:{u}_{{n}} \: \\ $$$$ \\ $$

Question Number 37817 Answers: 1 Comments: 1

calculate lim_(n→+∞) x^n (1−cos((π/x^n ))) with x from R and x≠0

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{x}^{{n}} \left(\mathrm{1}−{cos}\left(\frac{\pi}{{x}^{{n}} }\right)\right)\:{with}\:{x} \\ $$$${from}\:{R}\:{and}\:{x}\neq\mathrm{0} \\ $$

Question Number 37816 Answers: 0 Comments: 1

find lim_(x→0) ((ln(x+e^(sinx) ) −x^2 )/(sh(2x)))

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{{ln}\left({x}+{e}^{{sinx}} \right)\:−{x}^{\mathrm{2}} }{{sh}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 37815 Answers: 1 Comments: 1

let I = ∫_0 ^∞ e^(−x) cos^2 (π[x])dx and J = ∫_0 ^∞ e^(−x) sin^2 (π[x])dx 1) calculate I +J and I −J 2) find the values of I and J.

$${let}\:{I}\:\:=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} \:{cos}^{\mathrm{2}} \left(\pi\left[{x}\right]\right){dx}\:{and} \\ $$$${J}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}} \:{sin}^{\mathrm{2}} \left(\pi\left[{x}\right]\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}\:+{J}\:\:{and}\:{I}\:−{J} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:{I}\:{and}\:{J}. \\ $$

Question Number 37813 Answers: 1 Comments: 0

find A_n = ∫_(1/n) ^1 x(√x)arctan(x+(1/x))dx then calculate lim_(n→+∞) A_n .

$${find}\:{A}_{{n}} \:\:=\:\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \:\:{x}\sqrt{{x}}{arctan}\left({x}+\frac{\mathrm{1}}{{x}}\right){dx} \\ $$$${then}\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} . \\ $$

Question Number 37812 Answers: 1 Comments: 1

calculate ∫_0 ^∞ e^(−2x) sin{π[x]}dx .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{x}} {sin}\left\{\pi\left[{x}\right]\right\}{dx}\:. \\ $$

Question Number 37804 Answers: 2 Comments: 1

Question Number 37796 Answers: 0 Comments: 0

Find (dy/dx) if y= (x^2 + 1)^2 and y= (1−3x^2 )^5

$${Find}\:\frac{{dy}}{{dx}}\:{if}\: \\ $$$${y}=\:\left({x}^{\mathrm{2}} +\:\mathrm{1}\right)^{\mathrm{2}} \\ $$$${and}\:{y}=\:\left(\mathrm{1}−\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{5}} \\ $$

Question Number 37795 Answers: 1 Comments: 0

The side of a square is increasing at a rate of 0.1cms^(−1) . Find the rate of increase of the perimeter of the square when the length of side is 4cm.

$${The}\:{side}\:{of}\:{a}\:{square}\:{is}\:{increasing} \\ $$$${at}\:{a}\:{rate}\:{of}\:\mathrm{0}.\mathrm{1}{cms}^{−\mathrm{1}} .\:{Find}\:{the} \\ $$$${rate}\:{of}\:{increase}\:{of}\:{the}\:{perimeter} \\ $$$${of}\:{the}\:{square}\:{when}\:{the}\:{length}\:{of} \\ $$$${side}\:{is}\:\mathrm{4}{cm}. \\ $$

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