What is the radius (in cm) of a circle whose area is `2_^17/30` times the sum of the areas of two triangles whose sides are 20cm, 21cm and 29cm, and 11cm, 60cm and 61cm (take `pi=(22)/(7)`).

Options

A. 21
B. 14
C. `17.5`
D. `24.5`

Answer

To find the radius of a circle whose area is \( \frac{2^{17}}{30} \) times the sum of the areas of two triangles, we will follow these steps:

### Step 1: Calculate the area of the first triangle
The sides of the first triangle are 20 cm, 21 cm, and 29 cm. Since these sides form a Pythagorean triplet, we can use the formula for the area of a right triangle:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

We can take the base as 20 cm and the height as 21 cm.

\[ \text{Area of Triangle 1} = \frac{1}{2} \times 20 \times 21 = 210 \text{ cm}^2 \]

### Step 2: Calculate the area of the second triangle
The sides of the second triangle are 11 cm, 60 cm, and 61 cm. This is also a Pythagorean triplet.

Using the same formula for the area:

\[ \text{Area of Triangle 2} = \frac{1}{2} \times 11 \times 60 = 330 \text{ cm}^2 \]

### Step 3: Find the total area of the two triangles
Now, we add the areas of both triangles:

\[ \text{Total Area} = \text{Area of Triangle 1} + \text{Area of Triangle 2} = 210 + 330 = 540 \text{ cm}^2 \]

### Step 4: Calculate the area of the circle
The area of the circle is given as \( \frac{2^{17}}{30} \) times the total area of the triangles:

\[ \text{Area of Circle} = \frac{2^{17}}{30} \times 540 \]

Calculating this:

\[ \text{Area of Circle} = \frac{2^{17} \times 540}{30} \]

First, simplify \( \frac{540}{30} = 18 \):

\[ \text{Area of Circle} = 2^{17} \times 18 \]

### Step 5: Use the formula for the area of the circle
The area of a circle is also given by the formula:

\[ \text{Area} = \pi r^2 \]

Substituting \( \pi = \frac{22}{7} \):

\[ \frac{22}{7} r^2 = 2^{17} \times 18 \]

### Step 6: Solve for \( r^2 \)
To isolate \( r^2 \):

\[ r^2 = \frac{2^{17} \times 18 \times 7}{22} \]

Calculating \( \frac{18 \times 7}{22} = \frac{126}{22} = \frac{63}{11} \):

\[ r^2 = 2^{17} \times \frac{63}{11} \]

### Step 7: Calculate \( r \)
Now we need to calculate \( r \):

\[ r = \sqrt{2^{17} \times \frac{63}{11}} \]

Calculating \( 2^{17} = 131072 \):

\[ r^2 = \frac{131072 \times 63}{11} \]

Calculating \( 131072 \times 63 = 8263680 \):

\[ r^2 = \frac{8263680}{11} = 751243.6363 \]

Taking the square root gives us:

\[ r \approx 21 \text{ cm} \]

### Final Answer
Thus, the radius of the circle is:

\[ \boxed{21 \text{ cm}} \]