The number denoted by the area of a cicle is five times the number denoted by its circummference. Find the radius of the circle

Answer

To solve the problem, we need to find the radius of a circle where the area is five times the circumference. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
- The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]

2. **Set Up the Equation**:
- According to the problem, the area is five times the circumference. We can express this relationship mathematically:
\[ A = 5C \]
- Substituting the formulas for area and circumference into this equation gives:
\[ \pi r^2 = 5(2\pi r) \]

3. **Simplify the Equation**:
- Simplifying the right side:
\[ \pi r^2 = 10\pi r \]

4. **Divide Both Sides by \(\pi\)**:
- Since \(\pi\) is a common factor on both sides, we can divide both sides by \(\pi\) (assuming \(\pi \neq 0\)):
\[ r^2 = 10r \]

5. **Rearrange the Equation**:
- Rearranging gives us:
\[ r^2 - 10r = 0 \]

6. **Factor the Equation**:
- Factoring out \( r \):
\[ r(r - 10) = 0 \]

7. **Solve for \( r \)**:
- Setting each factor to zero gives:
\[ r = 0 \quad \text{or} \quad r - 10 = 0 \]
- Thus, \( r = 10 \) (we discard \( r = 0 \) as it does not make sense in this context).

8. **Conclusion**:
- The radius of the circle is:
\[ r = 10 \text{ units} \]