Mathematics: Corey, Laura, Paige and Jacob are baking a pumpkin pie. Each of them brings a different ingredient. When the pie is finished, 1/4 of the cinnamon that was brought is left, 1/3 of the nutmeg, 1/2 of the ginger, and 5/8 of the pumpkin remains. Corey s ingredient was used the most. Laura s ingredient was used the least. More of Paige s ingredient was used than Jacob s. Who brought the ginger?

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Corey, Laura, Paige and Jacob are baking a pumpkin

Ответ:

YoungLarQ

10.02.2022

Mathematics

1/4 cinnamon was left, so 3/4 was used.

1/3 nutmeg was left, so 2/3 was used

1/2 of ginger was left so 1/2 was left

5/8 of pumpkin was left so 3/8 was used.

Corey's was used the most, 3/4 is the largest used, so Corey had the cinnamon.

Laura's was used least, 3/8 is the smallest amount so Laura had the pumpkin.

2/3 is more then 1/2, so there was more ginger used than nutmeg.

More of Paige's was used, so Paige had the ginger.

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Ответ:

khuak877

25.08.2021

Mathematics

$\large\begin{array}{l}\\\\ \textsf{This is a complex number in }\mathsf{a+bi}\textsf{ form:}\\\\ \mathsf{z=-6+6\sqrt{3}\,i}\\\\ \textsf{where }\mathsf{a=-6~~and~~b=6\sqrt{3}.} \end{array}$

$\large\begin{array}{l}\\\\ \bullet~~\textsf{1}\mathsf{^{st}}\textsf{ step: Find the absolute value of z (complex modulus of z):}\\\\ \mathsf{|z|=\sqrt{a^2+b^2}}\\\\ \mathsf{|z|=\sqrt{(-6)^2+(6\sqrt{3})^2}}\\\\ \mathsf{|z|=\sqrt{36+36\cdot 3}}\\\\ \mathsf{|z|=\sqrt{36\cdot 108}}\\\\ \mathsf{|z|=\sqrt{144}}\\\\ \mathsf{|z|=12} \end{array}$

$\large\begin{array}{l}\bullet~~\textsf{2}\mathsf{^{nd}}\textsf{ step: Find the argument }\mathsf{\theta}\textsf{ of z:}\\\\ \mathsf{arg(z)=\theta}\\\\ \textsf{is an angle that satisfies the following conditions:}\\\\ \begin{cases} \mathsf{cos\,\theta=\dfrac{a}{|z|}=\dfrac{a}{\sqrt{a^2+b^2}}}\\\\ \mathsf{sin\,\theta=\dfrac{b}{|z|}=\dfrac{b}{\sqrt{a^2+b^2}}} \end{cases}\qquad\quad-\pi<\theta \le \pi \end{array}$

$\large\begin{array}{l}\textsf{For }\mathsf{z=-6+6\sqrt{3}\,i,}\\\\ \begin{cases} \mathsf{cos\,\theta=\dfrac{-6}{12}=-\,\dfrac{1}{2}}\\\\ \mathsf{sin\,\theta=\dfrac{6\sqrt{3}}{12}=\dfrac{\sqrt{3}}{2}} \end{cases}\\\\\\ \textsf{Since}\\\\ \mathsf{cos\,\theta<0~~and~~sin\,\theta0,}\\\\ \theta\textsf{ lies in the 2}\mathsf{^{nd}}\textsf{ quadrant.}\\\\\\\textsf{Therefore,}\\\\ \mathsf{\theta=\dfrac{2\pi}{3}~rad~~(120^\circ)} \end{array}$

$\large\begin{array}{l}\bullet~~\textsf{3}\mathsf{^{rd}}\textsf{ step: Express z in the trigonometric form:}\\\\ \begin{array}{rcl} \mathsf{z}&\!\!=\!\!&\mathsf{|z|\cdot (cos\,\theta+i\,sin\,\theta)}\\\\ \mathsf{-6+6\sqrt{3}\,i}&\!\!=\!\!&\mathsf{12\cdot \left(cos\, \dfrac{2\pi}{3}+i\,sin\,\dfrac{2\pi}{3}\right)\qquad\quad\checkmark} \end{array} \end{array}$

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$\large\textsf{I hope it helps.}$

Tags: complex number trigonometric form trig absolute value modulus argument angle

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