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 solve this problem, we need to calculate the area of both the bulls-eye (area A) and area B, and then determine the probability of hitting area B.
First, let's calculate the area of the bulls-eye (area A):
Area of bulls-eye (A) = πr^2
r = 4 inches (radius of bulls-eye)
A = π(4)^2
A = 16π square inches
Next, let's calculate the area of area B. Since area B consists of several concentric circles, we need to calculate the area of each circle and then subtract the area of the bulls-eye to get the area of B.
Area of first circle (B1):
r1 = r + 4 = 4 + 4 = 8 inches
Area of circle B1 = πr1^2
Area of circle B1 = π(8)^2
Area of circle B1 = 64π square inches
Area of second circle (B2):
r2 = r1 + 4 = 8 + 4 = 12 inches
Area of circle B2 = πr2^2
Area of circle B2 = π(12)^2
Area of circle B2 = 144π square inches
Now, we need to find the area of area B by subtracting the area of the bulls-eye from the total area of the circles:
Area of B = Area of circle B1 + Area of circle B2 - Area of bulls-eye
Area of B = 64π + 144π - 16π
Area of B = 192π square inches
Now that we have the area of B, we can calculate the probability of hitting area B randomly by dividing the area of B by the total area of the dartboard (which is the sum of the areas of A and B):
Total area of dartboard = Area of A + Area of B
Total area of dartboard = 16π + 192π
Total area of dartboard = 208π square inches
Probability of hitting area B = Area of B / Total area of dartboard
Probability of hitting area B = 192π / 208π
Probability of hitting area B = 0.9231
Therefore, the probability that a dart will hit in area B when thrown randomly onto the dartboard is approximately 0.9231 or 92.31%.