Answer:
The most correct option is;
(B) 958.2 ft.²
Step-by-step explanation:
From the question, the dimension of each square = 3 ft.²
Therefore, the length of the sides of the square = √3 ft.
Based on the above dimensions, the dimension of the small semicircle is found by counting the number of square sides ti subtends as follows;
The dimension of the diameter of the small semicircle = 10·√3
Radius of the small semicircle = Diameter/2 = 10·√3/2 = 5·√3
Area of the small semicircle = (π·r²)/2 = (π×(5·√3)²)/2 = 117.81 ft.²
Similarly;
The dimension of the diameter of the large semicircle = 10·√3 + 2 × 6 × √3
∴ The dimension of the diameter of the large semicircle = 22·√3
Radius of the large semicircle = Diameter/2 = 22·√3/2 = 11·√3
Area of the large semicircle = (π·r²)/2 = (π×(11·√3)²)/2 = 570.2 ft.²
Area of rectangle = 11·√3 × 17·√3 = 561
Area, A of large semicircle cutting into the rectangle is found as follows;
[tex]A_{(segment \, of \, semicircle)} = \frac{1}{4} \times (\theta - sin\theta) \times r^2[/tex]
Where:
[tex]\theta = 2\times tan^{-1}( \frac{The \, number \, of \, vertical \, squrare \, sides \ cut \, by \ the \ large \, semicircle}{The \, number \, of \, horizontal \, squrare \, sides \ cut \, by \ the \ large \, semicircle} )[/tex]
[tex]\therefore \theta = 2\times tan^{-1}( \frac{10\cdot \sqrt{3} }{5\cdot \sqrt{3}} ) = 2.214[/tex]
Hence;
[tex]A_{(segment \, of \, semicircle)} = \frac{1}{4} \times (2.214 - sin2.214) \times (11\cdot\sqrt{3} )^2 = 128.3 \, ft^2[/tex]
Therefore; t
The area covered by the pavers = 561 - 128.3 + 570.2 - 117.81 = 885.19 ft²
Therefor, the most correct option is (B) 958.2 ft.².
