An increase in the radius of a circle results in an increase of `6 (2)/(7)` cm in the circumference. Find the increase in the radius.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem
We need to find the increase in the radius of a circle when the circumference increases by \(6 \frac{2}{7}\) cm.

### Step 2: Write the formula for the circumference of a circle
The formula for the circumference \(C\) of a circle is given by:
\[ C = 2 \pi r \]
where \(r\) is the radius of the circle.

### Step 3: Define the variables
Let:
- \(r\) = original radius of the circle
- \(y\) = increase in the radius
- The new radius will be \(r + y\).

### Step 4: Write the new circumference
The new circumference after increasing the radius will be:
\[ C_{new} = 2 \pi (r + y) = 2 \pi r + 2 \pi y \]

### Step 5: Set up the equation
According to the problem, the increase in circumference is \(6 \frac{2}{7}\) cm. We can convert this mixed number into an improper fraction:
\[ 6 \frac{2}{7} = \frac{44}{7} \text{ cm} \]
Thus, we can write:
\[ C_{new} = C + \frac{44}{7} \]
Substituting the expressions for circumference, we have:
\[ 2 \pi r + 2 \pi y = 2 \pi r + \frac{44}{7} \]

### Step 6: Simplify the equation
Now, we can cancel \(2 \pi r\) from both sides:
\[ 2 \pi y = \frac{44}{7} \]

### Step 7: Solve for \(y\)
Now we need to isolate \(y\):
\[ y = \frac{44}{7} \div 2 \pi \]
Substituting \(\pi\) with \(\frac{22}{7}\):
\[ y = \frac{44}{7} \div 2 \times \frac{22}{7} = \frac{44}{7} \times \frac{7}{44} \]
The \(7\) and \(44\) cancel out:
\[ y = \frac{1}{2} \]

### Step 8: Conclusion
The increase in the radius \(y\) is:
\[ y = 1 \text{ cm} \]

### Final Answer
The increase in the radius is \(1\) cm.

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