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?

Answer 1

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%.