Let's solve the problem step by step.

Step 1: Understand the given information

• A smaller circle is inside a larger circle, with both sharing the same center point $M$.
• The circumference of the smaller circle is given as $6\pi$ inches.

Step 2: Find the radius of the smaller circle

The formula for the circumference of a circle is: $\text{Circumference} = 2\pi r$ where $r$ is the radius of the circle.

Given that the circumference of the smaller circle is $6\pi$: $2\pi r = 6\pi$ To find $r$, divide both sides by $2\pi$: $r = \frac{6\pi}{2\pi} = \frac{6}{2} = 3 \text{ inches}$

So, the radius of the smaller circle is $3$ inches.

Step 3: Determine the radius of the larger circle

The problem asks for the area of the larger circle, but first, we need to determine its radius.

Since the smaller circle is completely inside the larger circle and shares the same center, we can infer that the radius of the larger circle must be more than $3$ inches. However, without any additional information about the exact relationship between the sizes of the two circles, the problem may intend for us to assume that the diameter of the larger circle is equal to the diameter of the smaller circle plus the diameter of the larger circle.

Step 4: Area of the larger circle

Assuming that the larger circle's diameter is twice the radius of the smaller circle:

• The radius of the larger circle $R$ would be $2r = 2 \times 3 = 6$ inches.

The formula for the area of a circle is: $\text{Area} = \pi R^2$ Substitute $R = 6$ inches: $\text{Area of the larger circle} = \pi \times 6^2 = \pi \times 36 = 36\pi \text{ square inches}$

Answer

The area of the larger circle is $36\pi$ square inches.

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Do you want more details or have any other questions? Here are some related questions you might find interesting:

1. What if the smaller circle's radius was 4 inches instead of 3 inches, what would the area of the larger circle be?
2. How would the area of the larger circle change if the smaller circle was only tangent to the larger circle at one point?
3. What is the circumference of the larger circle?
4. What would be the area of the smaller circle?
5. How would the area change if the radius of the larger circle was 10 inches?
6. What if the radius of the larger circle was 9 inches, how would the problem change?
7. Can you explain how to derive the area formula for a circle?
8. How would you calculate the area if both circles were in a concentric configuration but the smaller circle was not perfectly centered?

Tip: In problems involving circles, it’s essential to correctly identify the radius, as it’s the key to solving for both area and circumference.