Question Number 181783 by mr W last updated on 30/Nov/22
$${one}\:{side}\:{of}\:{a}\:{triangle}\:{is}\:\mathrm{20}\:{cm}.\:{the} \\ $$$${other}\:{two}\:{sides}\:{are}\:{in}\:{ratio}\:\mathrm{1}:\mathrm{3}. \\ $$$$\left.\mathrm{1}\right)\:{what}\:{is}\:{the}\:{maximum}\:{area}\:{of}\:{the} \\ $$$${triangle},\:{if}\:{exists}? \\ $$$$\left.\mathrm{2}\right)\:{what}\:{is}\:{the}\:{maximun}\:{perimeter} \\ $$$${of}\:{the}\:{triangle},\:{if}\:{exists}? \\ $$
Answered by Frix last updated on 30/Nov/22
$$\mathrm{Let}\:\mathrm{the}\:\mathrm{sides}\:{x},\:\mathrm{3}{x} \\ $$$$\mathrm{5}<{x}<\mathrm{10} \\ $$$$\mathrm{The}\:\mathrm{collapsed}\:\mathrm{triangle}\:\mathrm{10},\mathrm{20},\mathrm{30}\:\mathrm{has}\:\mathrm{the} \\ $$$$\mathrm{maximum}\:\mathrm{perimeter}\:\mathrm{60} \\ $$$$\mathrm{The}\:\mathrm{area}\:\mathrm{is}\:{A}\left({x}\right)=\mathrm{2}\sqrt{−{x}^{\mathrm{4}} +\mathrm{125}{x}^{\mathrm{2}} −\mathrm{2500}} \\ $$$${A}'\left({x}\right)=\frac{−\mathrm{4}{x}\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{125}\right)}{{A}\left({x}\right)}=\mathrm{0}\wedge\mathrm{5}<{x}<\mathrm{10}\:\Rightarrow\:{x}=\frac{\mathrm{5}\sqrt{\mathrm{10}}}{\mathrm{2}} \\ $$$$\Rightarrow\:\mathrm{maximum}\:\mathrm{area}\:\mathrm{is}\:\mathrm{75} \\ $$
Commented by mr W last updated on 30/Nov/22
$${thanks}\:{sir}! \\ $$
Answered by MJS_new last updated on 30/Nov/22
$$\mathrm{generally}\:\mathrm{for}\:\mathrm{a}\:\mathrm{given}\:\mathrm{side}\:{a}\:\mathrm{and}\:\mathrm{a}\:\mathrm{ratio} \\ $$$${b}:{c}=\mathrm{1}:{v}\:\mathrm{with}\:{v}>\mathrm{1}\:\mathrm{we}\:\mathrm{get}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{area} \\ $$$$\frac{{a}^{\mathrm{2}} {v}}{\mathrm{2}\left({v}^{\mathrm{2}} −\mathrm{1}\right)}\:\mathrm{with}\:{b}=\frac{\sqrt{{v}^{\mathrm{2}} +\mathrm{1}}}{{v}^{\mathrm{2}} −\mathrm{1}}{a}\:\mathrm{and}\:{c}={vb} \\ $$