The area of circle is seven times the numerical value of its circumference. What is the circumference of the circle?

Options

A. 616 units
B. 132 units
C. 88 units
D. Cannot be determined

Explanation

To solve the problem, we need to find the circumference of a circle given that the area of the circle is seven times the numerical value of its circumference. Let's go through the steps:

### Step 1: Write down the formulas for area and circumference of a circle.
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:
\[ C = 2\pi r \]

### Step 2: Set up the equation based on the problem statement.
According to the problem, the area is seven times the circumference:
\[ A = 7C \]
Substituting the formulas for area and circumference into this equation gives:
\[ \pi r^2 = 7(2\pi r) \]

### Step 3: Simplify the equation.
We can simplify the equation by dividing both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ r^2 = 7(2r) \]
This simplifies to:
\[ r^2 = 14r \]

### Step 4: Rearrange the equation.
Rearranging the equation gives us:
\[ r^2 - 14r = 0 \]

### Step 5: Factor the equation.
Factoring out \( r \) from the equation:
\[ r(r - 14) = 0 \]
This gives us two solutions:
\[ r = 0 \quad \text{or} \quad r = 14 \]
Since the radius cannot be zero, we take \( r = 14 \).

### Step 6: Calculate the circumference.
Now that we have the radius, we can find the circumference using the formula:
\[ C = 2\pi r \]
Substituting \( r = 14 \):
\[ C = 2\pi(14) = 28\pi \]

### Step 7: Approximate the circumference.
Using \( \pi \approx \frac{22}{7} \):
\[ C = 28 \times \frac{22}{7} = 88 \text{ units} \]

Thus, the circumference of the circle is \( 88 \) units.