On the following dartboard, the radius of the bulls-eye (area A) is 4 inches. The radius of each concentric circle is 4 inches more than the circle inside it. If a person throws randomly onto the dartboard, what is the probability that the dart will hit in area B?
To find the probability that the dart will hit in area B, you need to calculate the area of the circle corresponding to area B and divide it by the total area of the dartboard.
Let's first find the radius of the circle corresponding to area B. The radius of area A (the bulls-eye) is 4 inches, so the radius of the circle corresponding to area B is 4 + 4 = 8 inches.
The area of a circle is calculated using the formula A = πr^2, where A is the area and r is the radius.
The area of the circle corresponding to area B is therefore A = π(8^2) = 64π square inches.
Now, let's find the total area of the dartboard. The total area consists of the sum of the areas of all the circles on the dartboard. The radius of the outermost circle is 4 + 3(4) = 16 inches.
The total area of the dartboard is the sum of the areas of the circles with radii 4, 8, 12, and 16 inches.
Total area = π(4^2) + π(8^2) + π(12^2) + π(16^2)
Total area = 16π + 64π + 144π + 256π
Total area = 480π square inches
Therefore, the probability that the dart will hit in area B is:
Probability = Area of B / Total area
Probability = 64π / 480π
Probability = 64/480
Probability = 4/30
Probability = 1/7
So, the probability that the dart will hit in area B is 1/7.