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Question-193635




Question Number 193635 by mr W last updated on 17/Jun/23
Commented by mr W last updated on 17/Jun/23
find the largest circle and the largest  square which you can completely  cover with three circular plates with  radius 1 respectively.
$${find}\:{the}\:{largest}\:{circle}\:{and}\:{the}\:{largest} \\ $$$${square}\:{which}\:{you}\:{can}\:{completely} \\ $$$${cover}\:{with}\:{three}\:{circular}\:{plates}\:{with} \\ $$$${radius}\:\mathrm{1}\:{respectively}. \\ $$
Answered by mr W last updated on 18/Jun/23
Commented by mr W last updated on 18/Jun/23
ΔABC=equilateral with side length                     s=2×r=2  R=(2/3)×(((√3)s)/2)=(s/( (√3)))=(2/( (√3)))≈1.155
$$\Delta{ABC}={equilateral}\:{with}\:{side}\:{length} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{s}=\mathrm{2}×{r}=\mathrm{2} \\ $$$${R}=\frac{\mathrm{2}}{\mathrm{3}}×\frac{\sqrt{\mathrm{3}}{s}}{\mathrm{2}}=\frac{{s}}{\:\sqrt{\mathrm{3}}}=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\approx\mathrm{1}.\mathrm{155} \\ $$
Answered by mr W last updated on 18/Jun/23
Commented by mr W last updated on 18/Jun/23
ΔABC=equilateral with side length                     s=2×r=2  (√(s^2 −a^2 ))=a−(s/( (√2)))  2a^2 −(√2)sa−(s^2 /2)=0  a^2 −(√2)a−1=0  ⇒a=(((√2)+(√6))/2)≈1.932
$$\Delta{ABC}={equilateral}\:{with}\:{side}\:{length} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{s}=\mathrm{2}×{r}=\mathrm{2} \\ $$$$\sqrt{{s}^{\mathrm{2}} −{a}^{\mathrm{2}} }={a}−\frac{{s}}{\:\sqrt{\mathrm{2}}} \\ $$$$\mathrm{2}{a}^{\mathrm{2}} −\sqrt{\mathrm{2}}{sa}−\frac{{s}^{\mathrm{2}} }{\mathrm{2}}=\mathrm{0} \\ $$$${a}^{\mathrm{2}} −\sqrt{\mathrm{2}}{a}−\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow{a}=\frac{\sqrt{\mathrm{2}}+\sqrt{\mathrm{6}}}{\mathrm{2}}\approx\mathrm{1}.\mathrm{932} \\ $$

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