+15 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?
+2668 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:
