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Question Number 66755 Answers: 4 Comments: 0

solve the simultaneous equations: 3x-y=9 x^2 -xy=4

$${solve}\:{the}\:{simultaneous}\:{equations}: \\ $$$$\mathrm{3}{x}-{y}=\mathrm{9} \\ $$$${x}^{\mathrm{2}} -{xy}=\mathrm{4} \\ $$

Question Number 66751 Answers: 1 Comments: 0

A rostum is made by cutting off the upper part of a cone along a plane parallel to the base at (2/3) up the height. What fraction of the volume of the cone does the rostum represent?

$${A}\:{rostum}\:{is}\:{made}\:{by}\:{cutting}\:{off} \\ $$$${the}\:{upper}\:{part}\:{of}\:{a}\:{cone}\:{along}\:{a} \\ $$$${plane}\:{parallel}\:{to}\:{the}\:{base}\:{at}\:\frac{\mathrm{2}}{\mathrm{3}}\:{up}\: \\ $$$${the}\:{height}.\:{What}\:{fraction}\:{of}\:{the}\: \\ $$$${volume}\:{of}\:{the}\:{cone}\:{does}\:{the} \\ $$$${rostum}\:{represent}? \\ $$

Question Number 66746 Answers: 2 Comments: 0

A point T divides a line AB internally in the ratio 5:2. Given that A is (-4,10) and B is (10,3), find the coordinates of T.

$${A}\:{point}\:{T}\:\:{divides}\:{a}\:{line}\:{AB}\:{internally}\:{in}\:{the}\:{ratio}\:\mathrm{5}:\mathrm{2}.\:{Given}\:{that}\:{A}\:{is}\:\left(-\mathrm{4},\mathrm{10}\right)\:{and}\:{B}\:{is}\:\left(\mathrm{10},\mathrm{3}\right),\:{find}\:{the}\:{coordinates}\:{of}\:{T}. \\ $$

Question Number 66743 Answers: 0 Comments: 4

Each month a store owner can spend at most $100,000 on PC′s and laptops. A PC costs the store owner $1000 and a laptop costs him $1500. Each PC is sold for a profit of $400 while a laptop is sold for a profit of $700. The store owner estimates that at least 15 PC′s but no more than 80 are sold each month. He also estimates that the number of laptops sold is at most half the PC′s. How many PC′s and how many laptops should be sold in order to maximize the profit?

$$\mathrm{Each}\:\mathrm{month}\:\mathrm{a}\:\mathrm{store}\:\mathrm{owner}\:\mathrm{can}\:\mathrm{spend}\:\mathrm{at} \\ $$$$\mathrm{most}\:\$\mathrm{100},\mathrm{000}\:\mathrm{on}\:\mathrm{PC}'\mathrm{s}\:\mathrm{and}\:\mathrm{laptops}.\:\mathrm{A} \\ $$$$\mathrm{PC}\:\mathrm{costs}\:\mathrm{the}\:\mathrm{store}\:\mathrm{owner}\:\$\mathrm{1000}\:\mathrm{and}\:\mathrm{a} \\ $$$$\mathrm{laptop}\:\mathrm{costs}\:\mathrm{him}\:\$\mathrm{1500}.\:\mathrm{Each}\:\mathrm{PC}\:\mathrm{is}\:\mathrm{sold} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\$\mathrm{400}\:\mathrm{while}\:\mathrm{a}\:\mathrm{laptop}\:\mathrm{is}\:\mathrm{sold} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\$\mathrm{700}.\:\mathrm{The}\:\mathrm{store}\:\mathrm{owner}\:\mathrm{estimates} \\ $$$$\mathrm{that}\:\mathrm{at}\:\mathrm{least}\:\mathrm{15}\:\mathrm{PC}'\mathrm{s}\:\mathrm{but}\:\mathrm{no}\:\mathrm{more}\:\mathrm{than} \\ $$$$\mathrm{80}\:\mathrm{are}\:\mathrm{sold}\:\mathrm{each}\:\mathrm{month}.\:\mathrm{He}\:\mathrm{also}\:\mathrm{estimates} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{laptops}\:\mathrm{sold}\:\mathrm{is}\:\mathrm{at}\:\mathrm{most} \\ $$$$\mathrm{half}\:\mathrm{the}\:\mathrm{PC}'\mathrm{s}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{PC}'\mathrm{s}\:\mathrm{and}\:\mathrm{how} \\ $$$$\mathrm{many}\:\mathrm{laptops}\:\mathrm{should}\:\mathrm{be}\:\mathrm{sold}\:\mathrm{in}\:\mathrm{order}\:\mathrm{to} \\ $$$$\mathrm{maximize}\:\mathrm{the}\:\mathrm{profit}? \\ $$

Question Number 66740 Answers: 0 Comments: 1

∫_( 0) ^( 1) (√(1−x+x^2 −x^3 )) dx=?

$$\underset{\:\:\mathrm{0}} {\overset{\:\:\:\:\:\:\:\mathrm{1}} {\int}}\sqrt{\mathrm{1}−\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:}\:\:\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 66739 Answers: 1 Comments: 0

find∫(√(dx ))

$${find}\int\sqrt{{dx}\:} \\ $$

Question Number 66734 Answers: 0 Comments: 0

Question Number 66731 Answers: 0 Comments: 1

y=x^2 −3x y=2x find area

$${y}={x}^{\mathrm{2}} −\mathrm{3}{x}\:\:\:\:\:{y}=\mathrm{2}{x}\:{find}\:{area} \\ $$

Question Number 66728 Answers: 2 Comments: 2

∫_1 ^∞ (1/(x(√(x^2 +1))))=?

$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{{x}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}=? \\ $$

Question Number 66727 Answers: 0 Comments: 0

let U_n =Σ_(k=0) ^n (1/(3k+1)) and H_n =Σ_(k=1) ^n (1/k) calculate U_n interms of H_n

$${let}\:\:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\mathrm{1}}{\mathrm{3}{k}+\mathrm{1}}\:\:\:{and}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$${calculate}\:{U}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$

Question Number 66718 Answers: 0 Comments: 3

Question Number 66715 Answers: 0 Comments: 4

if f(x)=((ln (x+(√(1+x^2 ))))/(√(1+x^2 ))) f^(−1) (x)=?

$${if}\:{f}\left({x}\right)=\frac{\mathrm{ln}\:\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)=? \\ $$

Question Number 66697 Answers: 1 Comments: 1

Question Number 66696 Answers: 2 Comments: 3

calculate lim_(x→0) ((arctan(1+x^3 )−(π/4))/(xsin(x^2 )))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{{arctan}\left(\mathrm{1}+{x}^{\mathrm{3}} \right)−\frac{\pi}{\mathrm{4}}}{{xsin}\left({x}^{\mathrm{2}} \right)} \\ $$

Question Number 66695 Answers: 1 Comments: 3

calculate ∫_0 ^∞ ((cos(arctanx))/(4+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({arctanx}\right)}{\mathrm{4}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 66694 Answers: 0 Comments: 3

let f(a) =∫_(−∞) ^(+∞) (dx/((x^4 +x^2 +a))) with a∈](1/4),+∞[ 1) calculate f(a) 2)find also g(a) =∫_(−∞) ^(+∞) (dx/((x^4 +x^2 +a)^2 )) 3) find the value of integrals ∫_0 ^∞ (dx/((x^4 +x^2 +3))) and ∫_0 ^∞ (dx/((x^4 +x^2 +1)^2 )) 4) developp f at integrserie.

$$\left.{let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} \:+{a}\right)}\:{with}\:{a}\in\right]\frac{\mathrm{1}}{\mathrm{4}},+\infty\left[\right. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{also}\:{g}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} +{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{3}\right)}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{4}\right)\:{developp}\:{f}\:{at}\:{integrserie}. \\ $$

Question Number 66693 Answers: 1 Comments: 1

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)^3 ))

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 66684 Answers: 1 Comments: 0