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?

fuvtest1 Feb 20, 2022

#2**0 **

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:

BuilderBoi Feb 20, 2022