Answer: The probability that a point chosen at random is in the blue region is approximately 0.96.
Explanation:
To find the probability that a point chosen at random is in the blue region, we need to compare the area of the blue region to the area of the entire circle.
The area of the circle is given by the formula A = πr^2, where r is the radius of the circle. Since we know the area of the circle is 314 square centimeters, we can solve for the radius:
A = πr^2
314 cm^2 = πr^2
r^2 = 100
r = 10 cm
The circle has a radius of 10 centimeters.
The square inside the circle has a side length of 2 centimeters, so its area is 2^2 = 4 square centimeters.
To find the area of the blue region, we need to subtract the area of the square from the area of the circle. Since the square is inscribed in the circle, the diameter of the circle is equal to the diagonal of the square. Using the Pythagorean theorem, we can find the diameter of the circle:
d^2 = 2s^2
d^2 = 2(2 cm)^2
d^2 = 8
d = sqrt(8) cm
The diameter of the circle is sqrt(8) centimeters.
The area of the circle is A = πr^2 = π(10 cm)^2 = 100π square centimeters.
The area of the blue region is the area of the circle minus the area of the square:
blue region = A - 4 cm^2
blue region = 100π - 4 square centimeters
The probability that a point chosen at random is in the blue region is therefore:
P(blue) = (blue region) / (total area)
P(blue) = (100π - 4) / (100π)
P(blue) ≈ 0.96
Therefore, the probability that a point chosen at random is in the blue region is approximately 0.96.