The diameter of a circle is 80 cm. The radii (in cm) of their concentric circles drawn in such a manner that the whole area is divided into four equal parts.

Options

A. `20 sqrt(2), 20 sqrt(3), 20`
B. `(10 sqrt(3))/(3), (10 sqrt(2))/(3), (10)/(3)`
C. `10 sqrt(3), 10 sqrt(2), 10`
D. 17, 14, 9

Answer

To solve the problem step by step, we will find the radii of the concentric circles that divide the area of the given circle into four equal parts.

### Step 1: Find the radius of the largest circle
The diameter of the circle is given as 80 cm. To find the radius (R) of the circle, we use the formula:

\[ R = \frac{D}{2} \]

Substituting the value of the diameter:

\[ R = \frac{80 \text{ cm}}{2} = 40 \text{ cm} \]

### Step 2: Calculate the area of the largest circle
The area (A) of a circle is given by the formula:

\[ A = \pi R^2 \]

Substituting the radius we found:

\[ A = \pi (40 \text{ cm})^2 = \pi \times 1600 \text{ cm}^2 \approx 3.14 \times 1600 \approx 5024 \text{ cm}^2 \]

### Step 3: Divide the area into four equal parts
Since the area is to be divided into four equal parts, we calculate the area of each part:

\[ \text{Area of each part} = \frac{5024 \text{ cm}^2}{4} = 1256 \text{ cm}^2 \]

### Step 4: Find the radius of the first concentric circle (r1)
The area of the first concentric circle (r1) is equal to the area of one part:

\[ \pi r_1^2 = 1256 \text{ cm}^2 \]

Solving for \(r_1\):

\[ r_1^2 = \frac{1256}{\pi} \approx \frac{1256}{3.14} \approx 400 \]

Taking the square root:

\[ r_1 = \sqrt{400} = 20 \text{ cm} \]

### Step 5: Find the radius of the second concentric circle (r2)
The area of the second concentric circle (r2) includes the area of the first circle plus the area of one part:

\[ \pi r_2^2 = 1256 \text{ cm}^2 + 1256 \text{ cm}^2 = 2512 \text{ cm}^2 \]

Solving for \(r_2\):

\[ r_2^2 = \frac{2512}{\pi} \approx \frac{2512}{3.14} \approx 800 \]

Taking the square root:

\[ r_2 = \sqrt{800} = 20\sqrt{2} \text{ cm} \]

### Step 6: Find the radius of the third concentric circle (r3)
The area of the third concentric circle (r3) includes the area of the first two circles plus one part:

\[ \pi r_3^2 = 1256 \text{ cm}^2 + 1256 \text{ cm}^2 + 1256 \text{ cm}^2 = 3768 \text{ cm}^2 \]

Solving for \(r_3\):

\[ r_3^2 = \frac{3768}{\pi} \approx \frac{3768}{3.14} \approx 1200 \]

Taking the square root:

\[ r_3 = \sqrt{1200} = 20\sqrt{3} \text{ cm} \]

### Final Result
The radii of the concentric circles are:
- \(r_1 = 20 \text{ cm}\)
- \(r_2 = 20\sqrt{2} \text{ cm}\)
- \(r_3 = 20\sqrt{3} \text{ cm}\)