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# The set of points in the plane that satisfy all four of the inequalities ​ is a polygon. What is its area?

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The set of points  in the plane that satisfy all four of the inequalities
\begin{align*} y &\ge 2x-2 \\ y &\le \frac{1}{4}x+4 \\ y &\ge 2\\ x &\ge 0\end{align*}
is a polygon. What is its area?

Feb 20, 2022

#1
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graphed it if it helps
https://www.desmos.com/calculator/ivyeezu2k0

Feb 20, 2022
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Alright, I'll take a shot.

It helps to inscribe the trapezoid within a rectangle. Using the boundaries:

$$x = 0$$

$$y = 2$$

$$y = 34/7$$

$$x = 24/7$$

The area of this rectangle is $${20\over7} \times {24\over7} = {480\over49}$$

The area of the trapezoid is the area of the rectangle minus both triangles on the sides of the trapezoid.

The first triangle has a height of $$6 \over7$$ and a width of $$24 \over7$$, making the area $$72 \over49$$.

The second triangle has a height of $$20 \over 7$$ and a width of $$10 \over 7$$, making the area $$100 \over 49$$

This means that the area of the triangle is: $${480\over49}-{72\over49}-{100\over49}=\color{brown}\boxed{308\over49}$$

Here image:

Feb 20, 2022
#3
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Also, this is my 200th answer!!!

BuilderBoi  Feb 20, 2022
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=^._.^=

catmg  Feb 20, 2022
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200 answers! That is impressive.  Congratulatons  BuilderBoi !

Melody  Feb 21, 2022