The diameters of two given circles are in the ratio `3 : 4` and the sum of the areas of the circles is equal to the area of a circle whose diameter measures 30m. Find the diameter of the given circles.

Answer

To solve the problem, we need to find the diameters of two circles given that their diameters are in the ratio of 3:4, and the sum of their areas equals the area of a circle with a diameter of 30 m.

### Step-by-Step Solution:

1. **Define the Diameters**:
Let the diameters of the two circles be \( d_1 \) and \( d_2 \). According to the problem, the ratio of the diameters is given as:
\[ \frac{d_1}{d_2} = \frac{3}{4} \]
This can be expressed as:
\[ d_1 = \frac{3}{4} d_2 \quad \text{(Equation 1)} \]

2. **Calculate the Area of the Circle with Diameter 30 m**:
The area \( A \) of a circle is given by the formula:
\[ A = \frac{\pi}{4} d^2 \]
For a circle with a diameter of 30 m, the area is:
\[ A = \frac{\pi}{4} (30)^2 = \frac{\pi}{4} \times 900 = 225\pi \quad \text{(Area of the circle with diameter 30 m)} \]

3. **Express the Areas of the Two Circles**:
The areas \( A_1 \) and \( A_2 \) of the two circles with diameters \( d_1 \) and \( d_2 \) can be expressed as:
\[ A_1 = \frac{\pi}{4} d_1^2 \]
\[ A_2 = \frac{\pi}{4} d_2^2 \]
The sum of the areas is:
\[ A_1 + A_2 = \frac{\pi}{4} d_1^2 + \frac{\pi}{4} d_2^2 = \frac{\pi}{4} (d_1^2 + d_2^2) \]

4. **Set Up the Equation**:
According to the problem, the sum of the areas of the two circles equals the area of the circle with a diameter of 30 m:
\[ \frac{\pi}{4} (d_1^2 + d_2^2) = 225\pi \]
Dividing both sides by \( \pi \) and \( \frac{1}{4} \):
\[ d_1^2 + d_2^2 = 900 \quad \text{(Equation 2)} \]

5. **Substitute Equation 1 into Equation 2**:
Substitute \( d_1 = \frac{3}{4} d_2 \) into Equation 2:
\[ \left(\frac{3}{4} d_2\right)^2 + d_2^2 = 900 \]
Simplifying this gives:
\[ \frac{9}{16} d_2^2 + d_2^2 = 900 \]
\[ \frac{9}{16} d_2^2 + \frac{16}{16} d_2^2 = 900 \]
\[ \frac{25}{16} d_2^2 = 900 \]

6. **Solve for \( d_2^2 \)**:
Multiply both sides by \( \frac{16}{25} \):
\[ d_2^2 = 900 \times \frac{16}{25} \]
\[ d_2^2 = 576 \]
Taking the square root:
\[ d_2 = 24 \text{ m} \]

7. **Find \( d_1 \)**:
Substitute \( d_2 = 24 \) m back into Equation 1:
\[ d_1 = \frac{3}{4} \times 24 = 18 \text{ m} \]

### Final Answer:
The diameters of the two circles are:
- \( d_1 = 18 \text{ m} \)
- \( d_2 = 24 \text{ m} \)