If the difference between the circumference and diameter of a circle is 30 cm, then the radius of the circle must be

Options

A. 6 cm
B. 7 cm
C. 5 cm
D. 8 cm

Answer

To solve the problem step by step, we will follow the logical reasoning based on the information given in the question.

### Step 1: Understand the problem
We need to find the radius of a circle given that the difference between its circumference and diameter is 30 cm.

### Step 2: Write down the formulas
- The formula for the circumference \( C \) of a circle is:
\[ C = 2\pi r \]
- The formula for the diameter \( D \) of a circle is:
\[ D = 2r \]

### Step 3: Set up the equation
According to the problem, the difference between the circumference and the diameter is 30 cm:
\[ C - D = 30 \]
Substituting the formulas for circumference and diameter:
\[ 2\pi r - 2r = 30 \]

### Step 4: Factor out common terms
We can factor out \( 2r \) from the left side of the equation:
\[ 2r(\pi - 1) = 30 \]

### Step 5: Solve for \( r \)
To isolate \( r \), we divide both sides by \( 2(\pi - 1) \):
\[ r = \frac{30}{2(\pi - 1)} \]
\[ r = \frac{15}{\pi - 1} \]

### Step 6: Calculate the value of \( r \)
Using the approximate value of \( \pi \approx 3.14 \):
\[ r = \frac{15}{3.14 - 1} = \frac{15}{2.14} \approx 7.01 \text{ cm} \]

### Conclusion
The radius of the circle is approximately \( 7 \text{ cm} \).

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