Two concentric circles are such that one circle has half the radius of the other. If a point is chosen at random inside the larger circle, the probability that it also lands outside the smaller circle is?
half the radius means a quarter of the area
3/4 of the larger circle is outside the smaller circle
To find the probability that a randomly chosen point inside the larger circle also lands outside the smaller circle, we need to determine the relative areas of the two circles.
Let's assume the radius of the larger circle is R units. According to the information given, the radius of the smaller circle would be R/2 units.
The area of a circle is given by the formula A = π * r^2, where A is the area and r is the radius.
Therefore, the area of the larger circle is A_large = π * R^2, and the area of the smaller circle is A_small = π * (R/2)^2.
Now we can determine the probability that a randomly chosen point inside the larger circle also lands outside the smaller circle, which is the ratio of the difference in areas to the area of the larger circle.
P = (A_large - A_small) / A_large
= (π * R^2 - π * (R/2)^2) / (π * R^2)
= (4πR^2 - πR^2) / (4πR^2)
= 3πR^2 / 4πR^2
= 3/4
Therefore, the probability that a randomly chosen point inside the larger circle also lands outside the smaller circle is 3/4, or 75%.
Let's assume the radius of the larger circle is R and the radius of the smaller circle is R/2.
To find the probability that a point chosen at random inside the larger circle also lands outside the smaller circle, we need to find the ratio of the area of the larger circle outside the smaller circle to the area of the larger circle.
The area of a circle can be calculated using the formula A = π * r^2, where r is the radius.
The area of the larger circle is A_larger = π * R^2.
The area of the smaller circle is A_smaller = π * (R/2)^2 = π * (R^2/4).
The area between the two circles is the difference between the area of the larger circle and the area of the smaller circle: A_between = A_larger - A_smaller.
So, A_between = π * R^2 - π * (R^2/4) = π * (4R^2/4 - R^2/4) = π * (3R^2/4).
The probability of landing in the area between the two circles is the ratio of A_between to A_larger: P = A_between / A_larger.
P = (π * (3R^2/4)) / (π * R^2) = (3R^2/4) / R^2 = 3/4.
Therefore, the probability that a point chosen at random inside the larger circle also lands outside the smaller circle is 3/4, or 75%.