Five concentric circles with radii 1, 2, 3, 4, and 5 are drawn as shown dividing the circle of radius 5 into five regions: a circle of radius 1 and 4 annuuli (rings). What is the probability that a point chosen randomly from within the circle of radius 5 lies in the annulus whose inner radius is 3 and who outer radius is 4?
the area of C5 = 25π
The area of the annulus between C3 and C4 is 16π-9π = 7π
so, p = 7/25
To find the probability that a point chosen randomly from within the circle of radius 5 lies in the annulus with inner radius 3 and outer radius 4, we need to find the ratio of the area of the annulus to the area of the entire circle.
Let's calculate the areas step-by-step:
1. The area of the entire circle with radius 5 can be found using the formula for the area of a circle: A = πr^2, where r is the radius.
So, the area of the circle is A1 = π(5^2).
2. The area of the inner circle with radius 3 is A2 = π(3^2).
3. To find the area of the annulus, subtract the area of the inner circle from the area of the entire circle: A_annulus = A1 - A2.
Now, let's calculate the areas:
A1 = π(5^2) = 25π (since 5^2 = 25)
A2 = π(3^2) = 9π (since 3^2 = 9)
A_annulus = A1 - A2 = 25π - 9π = 16π
The area ratio of the annulus to the entire circle is:
P = A_annulus / A1 = (16π) / (25π) = 16/25
Therefore, the probability that a point chosen randomly lies in the annulus is 16/25.
To find the probability that a randomly chosen point lies in the annulus between the circles with radii 3 and 4, you need to calculate the ratio of the area of the annulus to the total area of the circle with radius 5.
First, let's calculate the area of the annulus. The area of an annulus can be found by subtracting the area of the smaller circle (with radius 3) from the area of the larger circle (with radius 4).
The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius.
Area of the smaller circle (with radius 3):
A1 = π(3^2) = 9π
Area of the larger circle (with radius 4):
A2 = π(4^2) = 16π
Next, subtract the area of the smaller circle from the area of the larger circle to find the area of the annulus:
Area of the annulus = A2 - A1 = 16π - 9π = 7π
Now, let's find the area of the circle with radius 5:
A3 = π(5^2) = 25π
Finally, divide the area of the annulus by the total area of the circle to obtain the probability:
Probability = Area of the annulus / Area of the circle
Probability = (7π) / (25π) = 7/25
Therefore, the probability that a randomly chosen point lies in the annulus between the circles with radii 3 and 4 is 7/25.