Concentric circles have radii 1, 2, 3, 4, 5 and 6 cm. What is the probability that a randomly selected point falls in one of the shaded regions? Enter your answer as a common fraction, a/b. concentric circles. 6 concentric circles, every other circle is shaded, starting with a white circle and ending with a shaded ring.

To solve this problem, we need to calculate the probability that a randomly selected point falls in one of the shaded regions of the concentric circles. Here's the step-by-step process:

### Step 1: Areas of Each Circle
The area of a circle is given by the formula:
\[
A = \pi r^2
\]
where \(r\) is the radius of the circle.

For the six concentric circles, we calculate their areas using their respective radii:

- Area of circle with radius \(1\): \(A_1 = \pi(1^2) = \pi\)
- Area of circle with radius \(2\): \(A_2 = \pi(2^2) = 4\pi\)
- Area of circle with radius \(3\): \(A_3 = \pi(3^2) = 9\pi\)
- Area of circle with radius \(4\): \(A_4 = \pi(4^2) = 16\pi\)
- Area of circle with radius \(5\): \(A_5 = \pi(5^2) = 25\pi\)
- Area of circle with radius \(6\): \(A_6 = \pi(6^2) = 36\pi\)

### Step 2: Total Area of the Largest Circle
The largest circle has a radius of \(6\), so its total area is:
\[
\text{Total Area} = A_6 = 36\pi
\]

### Step 3: Identify Shaded Regions
From the description, every other circle is shaded, starting with the second one. This means the shaded regions correspond to the areas between:
- The circles with radii \(2\) and \(1\),
- The circles with radii \(4\) and \(3\),
- The circles with radii \(6\) and \(5\).

These regions are annular (ring-like), and their areas are calculated by subtracting the area of the smaller circle from the area of the larger circle in each pair.

- Shaded area between radii \(2\) and \(1\):
\[
\text{Area} = A_2 - A_1 = 4\pi - \pi = 3\pi
\]
- Shaded area between radii \(4\) and \(3\):
\[
\text{Area} = A_4 - A_3 = 16\pi - 9\pi = 7\pi
\]
- Shaded area between radii \(6\) and \(5\):
\[
\text{Area} = A_6 - A_5 = 36\pi - 25\pi = 11\pi
\]

### Step 4: Total Shaded Area
The total shaded area is the sum of the three shaded regions:
\[
\text{Total Shaded Area} = 3\pi + 7\pi + 11\pi = 21\pi
\]

### Step 5: Probability
The probability that a randomly selected point falls in one of the shaded regions is the ratio of the total shaded area to the total area of the largest circle. This is:
\[
\text{Probability} = \frac{\text{Total Shaded Area}}{\text{Total Area}} = \frac{21\pi}{36\pi} = \frac{21}{36} = \frac{7}{12}
\]

Thus, the probability is \( \boxed{\frac{7}{12}} \).

Would you like more details or have any further questions?

Here are some related questions:
1. What is the probability that a point falls in a non-shaded region?
2. How would the probability change if an additional circle with radius 7 cm is added, and it's shaded?
3. Can you generalize the formula for the probability of a shaded region for any number of concentric circles?
4. What is the area of the non-shaded regions?
5. How would the problem change if every other circle starting from the first is shaded?

**Tip**: To calculate areas between circles, subtract the area of the smaller circle from the area of the larger circle, treating them as annular rings.

Concentric circles have radii 1, 2, 3, 4, 5, and 6 cm. What is the probability that a randomly selected point falls in one of the shaded regions? Enter your answer as a common fraction, a/b. Six concentric circles, every other circle is shaded, starting with a white circle and ending with a shaded ring.

Find out how to calculate the probability that a randomly selected point falls in the shaded regions of six concentric circles with radii 1 to 6 cm. Learn the step-by-step process, including key geometric concepts and probability formulas. Ideal for students in Grades 8-10.

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