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. Calculate the percentage of material waste if no material is lost when two adjacent circles meet.

Correct result:

p =  35.1075 %
p₂ =  49.7345 %

Solution:

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

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

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