Cutting circles

From the square 1 m side we have to cut the circles with a radius of 10 cm. How many discs we cut and how many percent will be waste?

Result

n =  25
p =  21.46 %

Solution:

r=10 cm a=1 100=100 cm S1=a2=1002=10000 cm2 D=2 r=2 10=20 cm S2=D2=202=400 cm2 n=S1/S2=10000/400=25r=10 \ \text{cm} \ \\ a=1 \cdot \ 100=100 \ \text{cm} \ \\ S_{1}=a^2=100^2=10000 \ \text{cm}^2 \ \\ D=2 \cdot \ r=2 \cdot \ 10=20 \ \text{cm} \ \\ S_{2}=D^2=20^2=400 \ \text{cm}^2 \ \\ n=S_{1}/S_{2}=10000/400=25
S3=π r2=3.1416 102314.1593 cm2 p=100 (S1n S3)/S1=100 (1000025 314.1593)/1000021.460221.46%S_{3}=\pi \cdot \ r^2=3.1416 \cdot \ 10^2 \doteq 314.1593 \ \text{cm}^2 \ \\ p=100 \cdot \ (S_{1}-n \cdot \ S_{3})/S_{1}=100 \cdot \ (10000-25 \cdot \ 314.1593)/10000 \doteq 21.4602 \doteq 21.46 \%



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