asinus

Find the set of all possible values of a/b.

$a+b=1\\ a=1-b\\ b=1-a\\ \dfrac{a}{b}=\dfrac{a}{1-a}$

$\color{blue}Possibility_{\frac{a}{b}}= \{0<\mathbb R <∞\}$

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Let k be a positive real number and find  k, so that the line is tangent to the circle.

$y=\sqrt{k-x^2}\\ y=-x+3+k$

$\color{blue}k\notin \{+\mathbb{R}\}$

$y=\sqrt{k-x^2}\\ y=-x+3+k$

$y=\sqrt{{\color{blue}1.365}-x^2}\\ y=-x+3\color{blue}-1.365$

$\color{blue}k\in \{ \approx 1.365,\ \approx-1.365\}$   determined with graphics. 

If +k is replaced by -k in the problem, it is solvable. 

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Where is the circle sector shown??

What is the radius of the circle?

$A_{sector}=\dfrac{\pi r^2\alpha}{360°}\\ A_{sector}=\dfrac{60}{\pi}\\ r_{sector}=\sqrt{\dfrac{60\cdot 360°}{\pi \alpha}\cdot \dfrac{2\pi}{360°}}\\ \color{blue}r_{sector}=\sqrt{\dfrac{120}{\alpha }}$

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What is the area of rectangle ZWXY?

Let ZW be the baseline and WX be the height of the rectangle ZWXY, and ZW be x.

Then:

$\overline{WX}=\dfrac{x-2}{tan\ 60°}\\ \overline{AX}=\dfrac{tan\ 30°(x-2)}{tan\ 60°}=\dfrac{x-2}{3}\\ $

$\triangle ZAY\\ (x-\overline{AX})^2+\overline{WX}^2=4^2 \\ (x-\dfrac{x-2}{3})^2+(\dfrac{x-2}{tan\ 60°})^2=16\\ (\dfrac{2x}{3}+\dfrac{2}{3})^2+(\dfrac{x}{tan\ 60°}-\dfrac{2}{\tan\ 60°})^2=16\\ x\in \{-4,\dfrac{32}{7}\}\ \color{blue}WolframAlpha$

$A_{ZWXY}=x\cdot \dfrac{x-2}{tan\ 60°}=\dfrac{32}{7}\cdot \dfrac{\frac{32}{7}-2}{tan\ 60°}\\ \color{blue}A_{ZWXY}=6.7868$

The area of rectangle ZWXY is 6.7868.

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The number of solutions to
$x_1 + x_2 + x_3 + x_4 + x_5 \le 1$

in nonnegative integers:

$Each\ x_n\ can\ become\ =1, the\ remainings\ x_n\ become =0,\\ or\ all\ x_n\ become =0.$

$Example:\\ x_1\in \{1\}\\ x_{2}, x_{3},x_{4},x_{5} \in \{0\}\\ \color{blue}or\\ x_1,x_2, x_3,x_4,x_5 \in \{0\}$

$\color{blue}The\ number\ of\ solutions\ for\ nonnegative\ integers\ is\ 6.$

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$BC = 3, AC = 19\ and\ AB = 13.\\ BC<(AC-AB)\\ \color{blue }The\ requirements\ are\ pointless.$

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f(x)=12(x+2)/(x+1) grafic

At which integer-x is f(x) a integer?

There, f(x)=b.

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No, that's not helpful. It would be helpful if you explained your solution to us.

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Find the number of ordered pairs (a,b).

$\dfrac{{\color{blue}a} + 2}{a + 1} = \dfrac{\color{blue}b}{12}\\ $

$\dfrac{{\color{blue}1}+2}{1+1}=\dfrac{\color{blue}18}{12}\\ \dfrac{{\color{blue}3}+2}{3+1}=\dfrac{\color{blue}15}{12}\\ \dfrac{{\color{blue}5}+2}{5+1}=\dfrac{\color{blue}14}{12}\\ $

$\dfrac{{\color{blue}11}+2}{11+1}=\dfrac{\color{blue}13}{12}$

$\color{blue}(a,b)\in\{(1,18)\ ; (3,15)\ ; (5,14)\ ; (11,13)\}$

The number of solutions is 4.

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$Let\ \omega\ be\ a\ nonreal\ root\ of\ x^3 = 1.\\ Compute\ (1 - \omega + \omega^2)^6 + (1 + \omega - \omega^2)^6.$

$x^3=1\\ x=\sqrt[3]{1}\\ \color{blue}\sqrt[3]{1}\ is\ not\ a\ transcendental\ number.\ \sqrt[3]{1}=1 \\ \omega=1\\ (1 - 1 + 1^2)^6 + (1 + 1 - 1^2)^6=2$

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$0$

$Find\ the\ maximum\ value\ of\ \ 3x + 4y + 5z + x^3 + \dfrac{4x^2 y}{z} + \dfrac{z^5}{xy^2}.$

Plane geometry

$z^2=x^2+y^2\ \color{green }Pythagorean\ theorem\\ \color{blue}y=x\\ z^2=2x^2\\ \color{blue}z=x\sqrt{2}\\ x^2+y^2+z^2=1\\ x^2+x^2+2x^2=1\\$

$4x^2=1\\ \color{blue}x=\dfrac{1}{2}$

$f(x)=3x + 4y + 5z + x^3 + \dfrac{4x^2 y}{z} + \dfrac{z^5}{xy^2}\\ f(x)=3x+4x+5x\sqrt{2}+x^3+\dfrac{4x^3}{x\sqrt{2}}+\dfrac{(x\sqrt{2})^5}{x^3}\\ f(x)=x^3+6\sqrt{2}x^2+5\sqrt{2}x+7x\ \color{green}according\ to\ WolframAlpha\\ f(x_{0.5})=0.5^3+6\sqrt{2}(0.5)^2+5\sqrt{2}(0.5)+7(0.5)=9.28185\\ \dfrac{d}{dx}=3x^2+3⋅2^{\frac{5}{2}}x+5√2+7=0\\ {\color{blue}x\in \{-4.6477};-1.009\}$

$f(x)_{max}=(-4.6477)^3+6\sqrt{2}(-4.6477)^2+5\sqrt{2}(-4.6477)+7(-4.6477)\\ \color{blue}f(x)_{max}=17.4979$

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Find AB.

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$AB=\sqrt{409+(\frac{15}{sin(120°-\arctan{\frac{20}{3}})})^2-2*\sqrt{409}*\frac{15}{sin(120^°-\arctan{\frac{20}{3}})}*\cos{60°}}=22.4$

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The path to the farthest corner is \sqrt (2^2+1^2)= \sqrt 5.

Franklin reaches 100% of the cube surface.

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If K is the area of ​​the triangle, what should be determined if K is the area of ​​the triangle?

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