Question

An archery target with a radius of 61 cm has five scoring zones of equal widths. The colors of the zones are gold, red, blue, black, and white. The width of each colored zone is 12.2 cm and the radius of the central gold circle is also 12.2 cm. If an arrow hits the target at a random point, what is the probability that it hits the gold region? Round your answer to the nearest hundredth.

Answer 1

To solve this problem, we are going to need to compare the area of the central gold circle to the area of the overall target.

Given that the radius of the central gold circle is 12.2 cm and the area for a circle is
`\pi r^2`, the area of the central gold circle is:

`a = \pi (12.2)^2`

`a = 148.84\pi`

Similarly, we can find the area of the entire circular target given that the radius is 61 cm:

`A = \pi (61)^2`

`A = 3721\pi`

The probability of the arrow landing in the central gold circle would be the area of the central gold circle divided by the area of the entire target, or simply
`(a)/(A)`. This means that the probability is:

`\boxed{(148.84\pi)/(3721\pi) = 4\%}`

Answer 2

The area of each circle is proportional to the square of its radius. We don't actually need the dimensions of each circle, we only need to know that from inner to outer, the radii are 0.2, 0.4, 0.6, 0.8, and 1.0 of the outer radius. Then the areas of the circles, inner to outer, are 0.2² = .04, 0.4² = .16, 0.6² = .36, 0.8² = .64 and 1.0² = 1.0 of the total target area.

The are of each ring is the area of its circle less the area of the next smaller circle.

... outer ring (white) proportion is 1.00 - 0.64 = 0.36 of the target area

... black ring proportion is 0.64 - 0.36 = 0.28 of the target area

... blue ring proportion is 0.36 - 0.16 = 0.20 of the target area

... red ring proportion is 0.16 - 0.04 = 0.12 of the target area

... gold ring proportion is 0.04 - 0 = 0.04 of the target area

If the arrow hits the target at a random point, its probability of hitting the gold region is 0.04.