Find the radius of a circle whose circumference is sum of the circumferences of ten circles of radii 4 cm , 7 cm , 10 cm , 13 cm `"……."` etc.

Answer

To find the radius of a circle whose circumference is the sum of the circumferences of ten circles with given radii, we can follow these steps:

### Step 1: Identify the radii of the circles
The radii of the circles are given as follows:
- First circle: 4 cm
- Second circle: 7 cm
- Third circle: 10 cm
- Fourth circle: 13 cm
- Continuing this pattern, we can see that the radii increase by 3 cm each time.

### Step 2: Write down the radii of the ten circles
The radii of the first ten circles can be expressed as:
- 1st circle: 4 cm
- 2nd circle: 7 cm
- 3rd circle: 10 cm
- 4th circle: 13 cm
- 5th circle: 16 cm
- 6th circle: 19 cm
- 7th circle: 22 cm
- 8th circle: 25 cm
- 9th circle: 28 cm
- 10th circle: 31 cm

### Step 3: Calculate the circumference of each 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.

Now, we calculate the circumference for each of the ten circles:
1. Circumference of 1st circle: \( C_1 = 2\pi \times 4 = 8\pi \) cm
2. Circumference of 2nd circle: \( C_2 = 2\pi \times 7 = 14\pi \) cm
3. Circumference of 3rd circle: \( C_3 = 2\pi \times 10 = 20\pi \) cm
4. Circumference of 4th circle: \( C_4 = 2\pi \times 13 = 26\pi \) cm
5. Circumference of 5th circle: \( C_5 = 2\pi \times 16 = 32\pi \) cm
6. Circumference of 6th circle: \( C_6 = 2\pi \times 19 = 38\pi \) cm
7. Circumference of 7th circle: \( C_7 = 2\pi \times 22 = 44\pi \) cm
8. Circumference of 8th circle: \( C_8 = 2\pi \times 25 = 50\pi \) cm
9. Circumference of 9th circle: \( C_9 = 2\pi \times 28 = 56\pi \) cm
10. Circumference of 10th circle: \( C_{10} = 2\pi \times 31 = 62\pi \) cm

### Step 4: Sum the circumferences
Now we sum all the circumferences:
\[ \text{Total Circumference} = C_1 + C_2 + C_3 + C_4 + C_5 + C_6 + C_7 + C_8 + C_9 + C_{10} \]
\[ = 8\pi + 14\pi + 20\pi + 26\pi + 32\pi + 38\pi + 44\pi + 50\pi + 56\pi + 62\pi \]
\[ = (8 + 14 + 20 + 26 + 32 + 38 + 44 + 50 + 56 + 62)\pi \]

### Step 5: Calculate the sum of the coefficients
Calculating the sum of the coefficients:
\[ 8 + 14 + 20 + 26 + 32 + 38 + 44 + 50 + 56 + 62 = 350 \]
So, the total circumference is:
\[ \text{Total Circumference} = 350\pi \text{ cm} \]

### Step 6: Find the radius of the new circle
Let \( R \) be the radius of the new circle whose circumference is equal to the total circumference calculated above. Using the circumference formula:
\[ C = 2\pi R \]
Setting this equal to the total circumference:
\[ 2\pi R = 350\pi \]
Dividing both sides by \( 2\pi \):
\[ R = \frac{350\pi}{2\pi} = \frac{350}{2} = 175 \text{ cm} \]

### Final Answer
The radius of the circle is \( \mathbf{175 \, cm} \).