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# Geometry and Probability

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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.

Lightning  Mar 30, 2018
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### 1+0 Answers

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$$0\leq x \leq10 and 0\leq y \leq10$$

$$2x+5y \geq 20$$

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

CPhill  Mar 30, 2018

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