Sonia made two concentric circles with two pieces of rope. The distance between two circles is 6 cm. What is difference between lengths of 2 ropes used to make circles ?

Options

A. 12 cm
B. `12pi` cm
C. 36 cm
D. `36 pi` cm

Explanation

To solve the problem step by step, we will find the difference between the lengths of the two ropes used to make the concentric circles.

### Step 1: Define the Radii
Let the radius of the inner circle be \( R \) cm. Since the distance between the two circles is 6 cm, the radius of the outer circle will be:
\[ R + 6 \text{ cm} \]

### Step 2: Calculate the Length of the Outer Circle
The length (circumference) of the outer circle can be calculated using the formula for the circumference of a circle, which is \( 2\pi r \). Therefore, the length of the outer circle \( L_1 \) is:
\[ L_1 = 2\pi(R + 6) \]

### Step 3: Calculate the Length of the Inner Circle
Similarly, the length of the inner circle \( L_2 \) is:
\[ L_2 = 2\pi R \]

### Step 4: Find the Difference in Lengths
To find the difference between the lengths of the two ropes, we subtract the length of the inner circle from the length of the outer circle:
\[ \text{Difference} = L_1 - L_2 \]
Substituting the expressions we found:
\[ \text{Difference} = 2\pi(R + 6) - 2\pi R \]

### Step 5: Simplify the Expression
Now, simplifying the expression:
\[ \text{Difference} = 2\pi R + 12\pi - 2\pi R \]
The \( 2\pi R \) terms cancel out:
\[ \text{Difference} = 12\pi \]

### Conclusion
Thus, the difference between the lengths of the two ropes used to make the circles is:
\[ 12\pi \text{ cm} \]

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