Question Number 228162 by Hanuda354 last updated on 01/Apr/26
Commented by Danmola last updated on 01/Apr/26
$${nice}\:{problem} \\ $$
Commented by Hanuda354 last updated on 02/Apr/26
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{shaded}\:\mathrm{region}. \\ $$
Commented by fantastic2 last updated on 02/Apr/26
$${for}\:{minimum}\:\mathrm{6}\:{will}\:{be}\:{h} \\ $$$${area}=\mathrm{9}{x}^{\mathrm{2}} \\ $$
Answered by TonyCWX last updated on 01/Apr/26
$$\mathrm{There}\:\mathrm{is}\:\mathrm{not}\:\mathrm{enough}\:\mathrm{information}. \\ $$
Answered by Jyrgen last updated on 01/Apr/26
![I get shaded area = 180x+2304−1152(√3)−((6912(x+6(2−(√3))))/(x^2 +36)) and x≥−8+4(√3)+2(√(31−16(√3))) [≈2.55] if the triangle should lean to the right x<6(−2+(√6)−(√3)−(√2)) [≈4.60] the triangle is rectangular at x=6(2+(√3)) [≈22.39]](https://www.tinkutara.com/question/Q228182.png)
$${I}\:{get}\:{shaded}\:{area}\:= \\ $$$$\mathrm{180}{x}+\mathrm{2304}−\mathrm{1152}\sqrt{\mathrm{3}}−\frac{\mathrm{6912}\left({x}+\mathrm{6}\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)\right)}{{x}^{\mathrm{2}} +\mathrm{36}} \\ $$$${and}\:{x}\geqslant−\mathrm{8}+\mathrm{4}\sqrt{\mathrm{3}}+\mathrm{2}\sqrt{\mathrm{31}−\mathrm{16}\sqrt{\mathrm{3}}}\:\left[\approx\mathrm{2}.\mathrm{55}\right] \\ $$$${if}\:{the}\:{triangle}\:{should}\:{lean}\:{to}\:{the}\:{right} \\ $$$${x}<\mathrm{6}\left(−\mathrm{2}+\sqrt{\mathrm{6}}−\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}\right)\:\left[\approx\mathrm{4}.\mathrm{60}\right] \\ $$$${the}\:{triangle}\:{is}\:{rectangular}\:{at} \\ $$$${x}=\mathrm{6}\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)\:\left[\approx\mathrm{22}.\mathrm{39}\right] \\ $$
Commented by TonyCWX last updated on 01/Apr/26
$$\mathrm{My}\:\mathrm{original}\:\mathrm{assumption}\:\mathrm{was}\:\mathrm{that}\:\mathrm{75}°\:\mathrm{can}\:\mathrm{be}\:\mathrm{splitted}\:\mathrm{into}\:\mathrm{45}°\:\mathrm{and}\:\mathrm{30}°. \\ $$$$\mathrm{Then}\:\mathrm{there}'\mathrm{s}\:\mathrm{simply}\:\mathrm{not}\:\mathrm{enough}\:\mathrm{information}\:\mathrm{to}\:\mathrm{support}\:\mathrm{this}\:\mathrm{assumption}. \\ $$
Commented by Jyrgen last updated on 01/Apr/26
$${There}'{s}\:{not}\:{enough}\:{information}\:{to}\:{get} \\ $$$${one}\:{single}\:{value}\:{for}\:{the}\:{area}\:{but}\:{my} \\ $$$${solution}\:{is}\:{correct}.\:{Try}\:{with}\:{any}\:{value} \\ $$$${of}\:{x}. \\ $$