+1245 A point with coordinates $(x,\ y)$ is randomly selected such that $0\leq x \leq10$ and $0\leq y \leq10$. What is the probability that the coordinates of the point will satisfy $2x+5y \geq 20$? Express your answer as a common fraction.
+129852 \(0\leq x \leq10 and 0\leq y \leq10\)
\(2x+5y \geq 20 \)
The first two conditions set up a square that is 10 units on a side ....so the total area is 10^2 =
100 units^2 (1)
The area below the inequality 2x + 5y ≥ 20 [ but still within the square ] forms a triangle with a base of 10 and a height of 4
Its area = 10 * 4 / 2 = 20 units^2
Then the area above the inequality [ but still within the square is 100 - 20 ] = 80 units^2 (2)
So....the probability that the coordinates of a point satisfy all three conditions is
(2) / (1) = 80 / 100 = 4/5
Here's a graph : https://www.desmos.com/calculator/ome3jb01pa
