The radius of a circle whose circumference is equal to the sum of the circumferences of the two circles of diameters 36 cm and 20 cm is

Options

A. 56 cm
B. 42 cm
C. 28 cm
D. 16 cm

Explanation

To find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles with diameters of 36 cm and 20 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Circumference**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2 \pi r \]
where \( r \) is the radius of the circle.

2. **Calculate the Radius of the First Circle**:
The diameter of the first circle is given as 36 cm. The radius \( r_1 \) can be calculated as:
\[ r_1 = \frac{\text{Diameter}}{2} = \frac{36 \text{ cm}}{2} = 18 \text{ cm} \]

3. **Calculate the Radius of the Second Circle**:
The diameter of the second circle is given as 20 cm. The radius \( r_2 \) can be calculated as:
\[ r_2 = \frac{\text{Diameter}}{2} = \frac{20 \text{ cm}}{2} = 10 \text{ cm} \]

4. **Sum of the Circumferences of Both Circles**:
The circumferences of the two circles can be calculated as follows:
- Circumference of the first circle:
\[ C_1 = 2 \pi r_1 = 2 \pi (18) = 36 \pi \text{ cm} \]
- Circumference of the second circle:
\[ C_2 = 2 \pi r_2 = 2 \pi (10) = 20 \pi \text{ cm} \]
- Therefore, the sum of the circumferences is:
\[ C_1 + C_2 = 36 \pi + 20 \pi = 56 \pi \text{ cm} \]

5. **Set Up the Equation for the New Circle**:
We need to find the radius \( r \) of a new circle whose circumference is equal to the sum of the circumferences of the two circles:
\[ 2 \pi r = 56 \pi \]

6. **Solve for the Radius \( r \)**:
To find \( r \), we can divide both sides of the equation by \( 2 \pi \):
\[ r = \frac{56 \pi}{2 \pi} = \frac{56}{2} = 28 \text{ cm} \]

### Final Answer:
The radius of the circle is \( \boxed{28 \text{ cm}} \).