If the difference between the area of a regular hexagon and the area of a circle circumscribed it is `26.705m^(2)`. Find the radius of the circle. (Use `pi = 3.143, sqrt(3) = 1.732)`

Options

A. 4 cm
B. 7 cm
C. 11 cm
D. 9 cm

Answer

To solve the problem, we need to find the radius of a circle that circumscribes a regular hexagon, given that the difference between the area of the hexagon and the area of the circle is 26.705 m².

### Step-by-Step Solution:

1. **Understand the Areas Involved**:
- The area of a circle is given by the formula:
\[ \text{Area of Circle} = \pi R^2 \]
- The area of a regular hexagon can be calculated using the formula:
\[ \text{Area of Hexagon} = \frac{3\sqrt{3}}{2} s^2 \]
where \( s \) is the side length of the hexagon. For a regular hexagon inscribed in a circle of radius \( R \), the side length \( s \) is equal to \( R \).

2. **Express the Area of the Hexagon in Terms of R**:
- Since \( s = R \), we can rewrite the area of the hexagon:
\[ \text{Area of Hexagon} = \frac{3\sqrt{3}}{2} R^2 \]

3. **Set Up the Equation Based on the Given Difference**:
- According to the problem, the difference between the area of the circle and the area of the hexagon is given as:
\[ \pi R^2 - \frac{3\sqrt{3}}{2} R^2 = 26.705 \]

4. **Substitute the Values of π and √3**:
- Given \( \pi = 3.143 \) and \( \sqrt{3} = 1.732 \):
\[ 3.143 R^2 - \frac{3 \times 1.732}{2} R^2 = 26.705 \]

5. **Calculate the Coefficient of R²**:
- First, calculate \( \frac{3 \times 1.732}{2} \):
\[ \frac{3 \times 1.732}{2} = \frac{5.196}{2} = 2.598 \]
- Now substitute back into the equation:
\[ 3.143 R^2 - 2.598 R^2 = 26.705 \]
- Simplifying gives:
\[ (3.143 - 2.598) R^2 = 26.705 \]
\[ 0.545 R^2 = 26.705 \]

6. **Solve for R²**:
- Now divide both sides by 0.545:
\[ R^2 = \frac{26.705}{0.545} \approx 49 \]

7. **Find R**:
- Taking the square root of both sides:
\[ R = \sqrt{49} = 7 \]

8. **Conclusion**:
- The radius of the circle is:
\[ R = 7 \text{ meters} \]