Question Number 223538 by fantastic last updated on 28/Jul/25
$${x}+{y}=\mathrm{36} \\ $$$${xy}_{{max}} =?? \\ $$
Answered by Frix last updated on 28/Jul/25
$$\mathrm{Max}\:\mathrm{at}\:{x}={y}=\mathrm{18}\:\mathrm{because}\:\mathrm{out}\:\mathrm{of}\:\mathrm{all}\:\mathrm{rectangles} \\ $$$$\mathrm{with}\:\mathrm{given}\:\mathrm{circumference}\:\mathrm{the}\:\mathrm{square}\:\mathrm{has}\:\mathrm{the} \\ $$$$\mathrm{maximal}\:\mathrm{area} \\ $$
Commented by fantastic last updated on 28/Jul/25
$${thanks}\:{sir} \\ $$
Answered by Raphael254 last updated on 28/Jul/25
$$ \\ $$$${x}\:+\:{y}\:=\:\mathrm{36} \\ $$$${y}\:=\:\mathrm{36}\:−\:{x} \\ $$$$ \\ $$$${xy}\:=\:{x}\left(\mathrm{36}\:−\:{x}\right)\:=\:\mathrm{36}{x}\:−\:{x}^{\mathrm{2}} \\ $$$$ \\ $$$$−{x}^{\mathrm{2}} \:+\:\mathrm{36}\:{x}\:=\:\mathrm{0} \\ $$$$−{x}\left({x}\:−\:\mathrm{36}\right)\:=\:\mathrm{0} \\ $$$$ \\ $$$${x}\:=\:\mathrm{0}\:{or}\:{x}\:=\:\mathrm{36} \\ $$$$ \\ $$$${The}\:{vertex}\:{of}\:{the}\:{parabola}\:{will}\:{be}\:{at}\:{the}\:{average}\:{of}\:{the}\:{roots}:\:{x}\:=\:\frac{\mathrm{0}\:+\:\mathrm{36}}{\mathrm{2}}\:=\:\mathrm{18} \\ $$$$ \\ $$$${f}\left({x}\right)\:=\:−{x}^{\mathrm{2}} \:+\:\mathrm{36}{x} \\ $$$${f}\left(\mathrm{18}\right)\:=\:−\left(\mathrm{18}\right)^{\mathrm{2}} \:+\:\mathrm{36}\left(\mathrm{18}\right) \\ $$$${f}\left(\mathrm{18}\right)\:=\:−\mathrm{18}×\mathrm{18}\:+\:\mathrm{36}×\mathrm{18} \\ $$$${f}\left(\mathrm{18}\right)\:=\:\mathrm{18}×\mathrm{18} \\ $$$${f}\left(\mathrm{18}\right)\:=\:\mathrm{324} \\ $$