Metal washers

Metal washers with a diameter of 80 mm are cut from a strip of steel sheet with a width of 10 cm and a length of 2 m. When two adjacent circles meet, calculate the material waste percentage if no material is lost.

Correct answer:

p =  35.1075 %
p₂ =  49.7345 %

Step-by-step explanation:

$$\begin{gathered} a=10\text{cm} \\ D=80\text{mm}\to \text{cm}=80:10\text{cm}=8\text{cm} \\ r=D/2=8/2=4\text{cm} \\ {S}_1=\pi \cdot r^2=3.1416\cdot 4^2\doteq 50.2655{\text{cm}}^2 \\ (2r{)}^2=b^2+(a-2\cdot r{)}^2 \\ b=\sqrt{(2r{)}^2-(a-2\cdot r{)}^2}=\sqrt{(2\cdot 4{)}^2-(10-2\cdot 4{)}^2}=2\sqrt{15}\text{cm}\doteq 7.746\text{cm} \\ {S}_2=a\cdot b=10\cdot 7.746=20\sqrt{15}{\text{cm}}^2\doteq 77.4597{\text{cm}}^2 \\ p=\frac{S_2-S_1}{S_2}\cdot 100=\frac{77.4597-50.2655}{77.4597}\cdot 100=35.1075\% \end{gathered}$$

$$\begin{gathered} S_3=a^2=10^2=100{\text{cm}}^2 \\ {p}_2=100\cdot \frac{S_3-S_1}{S_3}=100\cdot \frac{100-50.2655}{100}=49.7345\% \end{gathered}$$

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