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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
avatar+15 
-1

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

 Feb 20, 2022
 #2
avatar+1384 
+2

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
avatar+1384 
+2

Also, this is my 200th answer!!!

BuilderBoi  Feb 20, 2022
 #4
avatar+2401 
0

Nice job on your 200th answer. :))

 

=^._.^=

catmg  Feb 20, 2022
 #5
avatar+117224 
+1

200 answers! That is impressive.  Congratulatons  BuilderBoi !

Melody  Feb 21, 2022

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