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Question Number 189899 by Mingma last updated on 23/Mar/23

Answered by HeferH last updated on 23/Mar/23

Commented by HeferH last updated on 24/Mar/23

(6−r)+(6(√3)−r)=12 6+6(√3)−12 = 2r r = ((6(√3)−6)/2) = 3(√3)−3 (x/(3(√3)−3))= ((6(√3)−(3(√3)−3))/( (√((3(√3)−3)^2 +36)))) x = (((3(√3))^2 −9)/( (√(72−18(√3))))) = ((18)/( (√(8∙9−2∙9(√3)))))=((18)/(3(√(8−2(√3))))) x = (6/( (√(8−2(√3))))) ≈ 2.81

$$\left(\mathrm{6}−\mathrm{r}\right)+\left(\mathrm{6}\sqrt{\mathrm{3}}−\mathrm{r}\right)=\mathrm{12} \\ $$$$\:\mathrm{6}+\mathrm{6}\sqrt{\mathrm{3}}−\mathrm{12}\:=\:\mathrm{2r} \\ $$$$\:\mathrm{r}\:=\:\frac{\mathrm{6}\sqrt{\mathrm{3}}−\mathrm{6}}{\mathrm{2}}\:=\:\mathrm{3}\sqrt{\mathrm{3}}−\mathrm{3} \\ $$$$\:\frac{\mathrm{x}}{\mathrm{3}\sqrt{\mathrm{3}}−\mathrm{3}}=\:\frac{\mathrm{6}\sqrt{\mathrm{3}}−\left(\mathrm{3}\sqrt{\mathrm{3}}−\mathrm{3}\right)}{\:\sqrt{\left(\mathrm{3}\sqrt{\mathrm{3}}−\mathrm{3}\right)^{\mathrm{2}} +\mathrm{36}}} \\ $$$$\:\mathrm{x}\:=\:\frac{\left(\mathrm{3}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} −\mathrm{9}}{\:\sqrt{\mathrm{72}−\mathrm{18}\sqrt{\mathrm{3}}}}\:=\:\frac{\mathrm{18}}{\:\sqrt{\mathrm{8}\centerdot\mathrm{9}−\mathrm{2}\centerdot\mathrm{9}\sqrt{\mathrm{3}}}}=\frac{\mathrm{18}}{\mathrm{3}\sqrt{\mathrm{8}−\mathrm{2}\sqrt{\mathrm{3}}}} \\ $$$$\:\mathrm{x}\:=\:\frac{\mathrm{6}}{\:\sqrt{\mathrm{8}−\mathrm{2}\sqrt{\mathrm{3}}}}\:\approx\:\mathrm{2}.\mathrm{81} \\ $$

Commented by Rupesh123 last updated on 24/Mar/23

Excellent!

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