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?

Lilliam0216 Mar 3, 2024

#1**+1 **

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

CPhill Mar 3, 2024