a) The area of the shaded region is 50π.
b) [tex]\frac{Area of the shaded region}{area of square} = \frac{\pi}{8} .[/tex]
Step-by-step explanation:
Step 1:
The area of a circle [tex]= \pi r^{2}.[/tex]
The area of a semi-circle [tex]= \frac{\pi r^{2} }{2} .[/tex]
The area of a quarter-circle [tex]= \frac{\pi r^{2} }{4} .[/tex]
The shaded region's area is obtained by subtracting the area of the semi-circle with a radius of 10 cm from the area of a quarter-circle with a radius of 20 cm.
The area of the semi-circle with a radius of 10 cm [tex]= \frac{\pi (10^{2}) }{2} .[/tex]
The area of the quarter-circle with a radius of 20 cm [tex]= \frac{\pi (20^{2}) }{4} .[/tex]
Step 2:
The area of the shaded region is obtained by subtracting these areas.
The area of the shaded region [tex]= \frac{\pi (20^{2}) }{4} - \frac{\pi (10^{2}) }{2} .[/tex]
[tex]\frac{\pi (20^{2}) }{4} - \frac{\pi (10^{2}) }{2} = \frac{\pi}{2} (\frac{20^{2} }{2} -10^{2} ).[/tex]
[tex]\frac{\pi}{2} (\frac{20^{2} }{2} -10^{2} ) = \frac{\pi}{2} (\frac{400}{2} -100 ) = \frac{\pi}{2} (100).[/tex]
[tex]\frac{\pi}{2} (100) = 50 \pi.[/tex]
The area of the shaded region is 50π square units.
Step 3:
The area of a square is the square of the side length.
The area of the square [tex]= 20^{2} = 400[/tex] square cm.
b) [tex]\frac{Area of the shaded region}{area of square} =\frac{50\pi}{400} =\frac{\pi}{8} .[/tex]
