Question-212928

Question Number 212928 by efronzo1 last updated on 27/Oct/24

Commented by mr W last updated on 27/Oct/24

does the question mean msximum or minimum? if C and D must lie between B and E, then there is only min(x+y), but no max(x+y). if C and D may also lie at B and E, then max(x+y)=max((√(6^2 +6^2 ))+2, 6+(√(2^2 +6^2 ))) =2+6(√2) if the question means min(x+y), answer see below.

$${does}\:{the}\:{question}\:{mean}\:{msximum} \\ $$$${or}\:{minimum}?\:{if}\:{C}\:{and}\:{D}\:{must}\:{lie} \\ $$$${between}\:{B}\:{and}\:{E},\:{then}\:{there}\:{is}\:{only} \\ $$$${min}\left({x}+{y}\right),\:{but}\:{no}\:{max}\left({x}+{y}\right).\:{if}\:{C} \\ $$$${and}\:{D}\:{may}\:{also}\:{lie}\:{at}\:{B}\:{and}\:{E},\:{then} \\ $$$${max}\left({x}+{y}\right)={max}\left(\sqrt{\mathrm{6}^{\mathrm{2}} +\mathrm{6}^{\mathrm{2}} }+\mathrm{2},\:\mathrm{6}+\sqrt{\mathrm{2}^{\mathrm{2}} +\mathrm{6}^{\mathrm{2}} }\right) \\ $$$$=\mathrm{2}+\mathrm{6}\sqrt{\mathrm{2}} \\ $$$${if}\:{the}\:{question}\:{means}\:{min}\left({x}+{y}\right), \\ $$$${answer}\:{see}\:{below}. \\ $$

Answered by golsendro last updated on 27/Oct/24

x=(√(36+a^2 )) y=(√(4+(6−a)^2 )) Let x+y = f(a)=(√(36+a^2 )) + (√(4+(6−a)^2 )) f ′(a)= (a/( (√(36+a^2 )))) −(((6−a))/( (√(40−12a+a^2 ))))=0 a^2 (40−12a+a^2 )= (36+a^2 )(36−12a+a^2 )

$$\:\:\mathrm{x}=\sqrt{\mathrm{36}+\mathrm{a}^{\mathrm{2}} } \\ $$$$\:\:\mathrm{y}=\sqrt{\mathrm{4}+\left(\mathrm{6}−\mathrm{a}\right)^{\mathrm{2}} } \\ $$$$\:\:\mathrm{Let}\:\mathrm{x}+\mathrm{y}\:=\:\mathrm{f}\left(\mathrm{a}\right)=\sqrt{\mathrm{36}+\mathrm{a}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{4}+\left(\mathrm{6}−\mathrm{a}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\mathrm{f}\:'\left(\mathrm{a}\right)=\:\frac{\mathrm{a}}{\:\sqrt{\mathrm{36}+\mathrm{a}^{\mathrm{2}} }}\:−\frac{\left(\mathrm{6}−\mathrm{a}\right)}{\:\sqrt{\mathrm{40}−\mathrm{12a}+\mathrm{a}^{\mathrm{2}} }}=\mathrm{0} \\ $$$$\:\:\:\mathrm{a}^{\mathrm{2}} \:\left(\mathrm{40}−\mathrm{12a}+\mathrm{a}^{\mathrm{2}} \right)=\:\left(\mathrm{36}+\mathrm{a}^{\mathrm{2}} \right)\left(\mathrm{36}−\mathrm{12a}+\mathrm{a}^{\mathrm{2}} \right) \\ $$$$ \\ $$

Answered by mr W last updated on 27/Oct/24