+311 The set of points \((x, y)\) 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?
+129771 See the graph below
The polygon is bounded by (0,2) , (0,4), (2,2) and the point determined by the intersection of the lines
y= 2x - 2
y = (1/4)x + 4
To find the x intersection of these lines
2x -2 = (1/4)x + 4
(7/4)x = 6
x = 24/7
y = 2(24/7) - 2 = 34/7
The last boundary point is (24/7,34/7)
We can split this into two areas
The first is a triangle with a base = 3 and a height = (34/7 - 4) = 6/7
The area of this = (1/2)(3)(6/7) = 9/7
The second area is a trapezoid with bases = 2,3 and a height = 2
Its area = (1/2)(2)(2 + 3) = 5
The total area = 5 + 9/7 = 44/7
