The sum of diameter of two circles is 14 cm and the difference of their circumferences is 8 cm. Find the circumference of two circles

Answer

To solve the problem, we need to find the circumferences of two circles given the sum of their diameters and the difference of their circumferences.

Let's denote:
- The diameter of the first circle as \( d_1 \)
- The diameter of the second circle as \( d_2 \)
- The circumference of the first circle as \( C_1 \)
- The circumference of the second circle as \( C_2 \)

### Step 1: Set up the equations
From the problem statement, we have two equations:
1. The sum of the diameters:
\[ d_1 + d_2 = 14 \quad \text{(1)} \]
2. The difference of the circumferences:
\[ C_2 - C_1 = 8 \quad \text{(2)} \]

### Step 2: Express the circumferences in terms of the diameters
The circumference of a circle is given by the formula:
\[ C = \pi \times d \]
Thus, we can express \( C_1 \) and \( C_2 \) as:
\[ C_1 = \pi d_1 \quad \text{(3)} \]
\[ C_2 = \pi d_2 \quad \text{(4)} \]

### Step 3: Substitute equations (3) and (4) into equation (2)
Substituting \( C_1 \) and \( C_2 \) from equations (3) and (4) into equation (2):
\[ \pi d_2 - \pi d_1 = 8 \]
Factoring out \( \pi \):
\[ \pi (d_2 - d_1) = 8 \]
Dividing both sides by \( \pi \):
\[ d_2 - d_1 = \frac{8}{\pi} \quad \text{(5)} \]

### Step 4: Solve the system of equations (1) and (5)
Now we have a system of equations:
1. \( d_1 + d_2 = 14 \) (equation 1)
2. \( d_2 - d_1 = \frac{8}{\pi} \) (equation 5)

We can solve for \( d_2 \) from equation (5):
\[ d_2 = d_1 + \frac{8}{\pi} \]
Substituting this expression for \( d_2 \) into equation (1):
\[ d_1 + \left(d_1 + \frac{8}{\pi}\right) = 14 \]
Combining like terms:
\[ 2d_1 + \frac{8}{\pi} = 14 \]
Subtract \( \frac{8}{\pi} \) from both sides:
\[ 2d_1 = 14 - \frac{8}{\pi} \]
Dividing by 2:
\[ d_1 = 7 - \frac{4}{\pi} \quad \text{(6)} \]

### Step 5: Find \( d_2 \)
Now substitute \( d_1 \) back into equation (1) to find \( d_2 \):
\[ d_2 = 14 - d_1 = 14 - \left(7 - \frac{4}{\pi}\right) \]
Simplifying:
\[ d_2 = 7 + \frac{4}{\pi} \quad \text{(7)} \]

### Step 6: Calculate the circumferences
Now we can find the circumferences \( C_1 \) and \( C_2 \):
Using equation (3):
\[ C_1 = \pi d_1 = \pi \left(7 - \frac{4}{\pi}\right) = 7\pi - 4 \]
Using equation (4):
\[ C_2 = \pi d_2 = \pi \left(7 + \frac{4}{\pi}\right) = 7\pi + 4 \]

### Final Result
Thus, the circumferences of the two circles are:
\[ C_1 = 7\pi - 4 \quad \text{and} \quad C_2 = 7\pi + 4 \]