a rectangle has a length of 12 units and an unknown width, x. using p to represent the perimeter and a to represent the area, wr

ite an equation for the perimeter of the rectangle and the area of the rectangle

Mathematics

1 answer:

Amanda [17]1 year ago

4 0

$\begin{gathered} L\text{ = 12} \\ \text{Breadth / width = x} \\ \text{Area, a = L }\times\text{ B} \\ \text{ = 12 }\times\text{ a } \\ \text{ = 12a} \\ \text{Perimeter, P: = 2(L + B)} \\ \text{ = 2(12 + x)} \\ \text{ = 24 + 2x} \\ \end{gathered}$

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Solve the equation on the interval [0,2π]

WARRIOR [948]

$\bf 16sin^5(x)+2sin(x)=12sin^3(x)
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16sin^5(x)+2sin(x)-12sin^3(x)=0
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\stackrel{common~factor}{2sin(x)}[8sin^4(x)+1-6sin^2(x)]=0\\\\
-------------------------------\\\\
2sin(x)=0\implies sin(x)=0\implies \measuredangle x=sin^{-1}(0)\implies \measuredangle x=
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0\\
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$\bf 8sin^4(x)+1-6sin^2(x)=0\implies 8sin^4(x)-6sin^2(x)+1=0$

now, this is a quadratic equation, but the roots do not come out as integers, however it does have them, the discriminant, b² - 4ac, is positive, so it has 2 roots, so we'll plug it in the quadratic formula,

$\bf 8sin^4(x)-6sin^2(x)+1=0\implies 8[~[sin(x)]^2~]^2-6[sin(x)]^2+1=0
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~~~~~~~~~~~~\textit{quadratic formula}
\\\\
\begin{array}{lcccl}
& 8 sin^4& -6 sin^2(x)& +1\\
&\uparrow &\uparrow &\uparrow \\
&a&b&c
\end{array} 
\qquad \qquad 
sin(x)= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}
\\\\\\
sin(x)=\cfrac{-(-6)\pm\sqrt{(-6)^2-4(8)(1)}}{2(8)}\implies sin(x)=\cfrac{6\pm\sqrt{4}}{16}
\\\\\\
sin(x)=\cfrac{6\pm 2}{16}\implies sin(x)=
\begin{cases}
\frac{1}{2}\\\\
\frac{1}{4}
\end{cases}$

$\bf \measuredangle x=
\begin{cases}
sin^{-1}\left( \frac{1}{2} \right)
sin^{-1}\left( \frac{1}{4} \right)
\end{cases}\implies \measuredangle x=
\begin{cases}
\frac{\pi }{6}~,~\frac{5\pi }{6}\\
----------\\
\approx~0.252680~radians\\
\qquad or\\
\approx~14.47751~de grees\\
----------\\
\pi -0.252680\\
\approx 2.88891~radians\\
\qquad or\\
180-14.47751\\
\approx 165.52249~de grees
\end{cases}$

3 0

3 years ago

STALIN [3.7K]

They all have either a common factor, common multiple, LCF, GCF, LCM, or GCM. The G and L stand for greater and least.

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3 years ago

Read 2 more answers

PLEASE HELP! THANK YOU! :D

Ostrovityanka [42]

I think it’s C. Sorry if I’m incorrect

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3 years ago