To find the area of the courtyard that is in the shape of a rectangle and a half-circle, we can break it down into two parts: the rectangular area and the area of the half-circle.
-
Calculate the area of the rectangle: The formula for the area of a rectangle is: \[ \text{Area of rectangle} = \text{length} \times \text{width} \] Given dimensions:
- Length = 100 m
- Width = 50 m
\[ \text{Area of rectangle} = 100 , \text{m} \times 50 , \text{m} = 5000 , \text{m}^2 \]
-
Calculate the area of the half-circle: The formula for the area of a circle is: \[ \text{Area of circle} = \pi r^2 \] Since we have a half-circle, we need to divide this area by 2. The radius of the half-circle is half the width of the rectangle, which is: \[ r = \frac{50 , \text{m}}{2} = 25 , \text{m} \]
So, the area of the half-circle is: \[ \text{Area of half-circle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (25 , \text{m})^2 \] \[ = \frac{1}{2} \pi (625 , \text{m}^2) \approx \frac{1}{2} \times 3.14 \times 625 \approx 981.25 , \text{m}^2 \]
-
Total area of the courtyard: Now, we add both areas together: \[ \text{Total Area} = \text{Area of rectangle} + \text{Area of half-circle} \] \[ \text{Total Area} = 5000 , \text{m}^2 + 981.25 , \text{m}^2 \approx 5981.25 , \text{m}^2 \]
Rounding this value gives approximately \( 5982 , \text{m}^2 \).
Based on the options provided, the closest answer would be: C. 5,982 m²