+0  
 
0
3
1
avatar+96 

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?

 Mar 3, 2024
 #1
avatar+128448 
+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

 

 

cool cool cool

 Mar 3, 2024

0 Online Users