If a point in the square is randomly selected and there are nine congruent circles outlined there, there is a 21.5% chance that the point is not within a circle.
It is given to us that -
Nine congruent circles are inscribed in a square
The square has a side length of 126
We have to find out the probability that the point is not in a circle, if a point in the square is chosen at random.
It is known that the square has a side length of 126.
=> Area of the square = [tex](Length)^{2}[/tex]
=> Area of the square = [tex]126^{2}[/tex]
=> Area of the square = 15876 ------ (1)
It is also known to us that nine congruent circles are inscribed in the square that has a side length of 126.
=> Diameter of each circle = 126/3
=> Diameter of each circle = 42
=> Radius of each circle = 21 ------ (2)
We know that the area of a circle is given as -
Area of circle = [tex]\pi r^{2}[/tex] ------ (3)
where,
r = radius of the circle
Substituting the value of r from equation (2) in equation (1), we have
Area of circle = [tex]\pi r^{2}[/tex]
=> Area of circle = [tex]\pi (21)^{2}[/tex]
=> Area of circle = 1385.44
=> Area of 9 circles = 12468.96 ----- (4)
Now, we can say that -
Area not in a circle = Area of square - Area of 9 circles
=> Area not in a circle = 15876 - 12468.96 [From equation (1) and (4)]
=> Area not in a circle = 3407.04 ------ (5)
We know that the probability of a outcome is given as -
Probability = Number of favorable outcomes/Total number of outcomes
So, the probability that the point is not in a circle can be calculated as -
Probability = Area not in a circle/Area of square
=> Probability = 3407.04/15876
=> Probability = 0.215
=> Probability = 21.5%
Thus, if there are nine congruent circles inscribed in a square and a point in the square is chosen at random, the probability that the point is not in a circle is 21.5%.
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