Question Number 59100 by ajfour last updated on 04/May/19
Commented by ajfour last updated on 04/May/19
$$\mathrm{A}\:\mathrm{is}\:\mathrm{on}\:\mathrm{the}\:\mathrm{rim}\:\mathrm{of}\:\mathrm{a}\:\mathrm{hemisphere}, \\ $$$$\mathrm{radius}\:\mathrm{R}.\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{on}\:\mathrm{the}\:\mathrm{hemisphere} \\ $$$$\mathrm{surface};\:\mathrm{find}\:\mathrm{maximum}\:\mathrm{area}\:\bigtriangleup\mathrm{ABC}. \\ $$
Commented by MJS last updated on 04/May/19
$$\mathrm{I}\:\mathrm{think}\:{ABC}\:\mathrm{must}\:\mathrm{form}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle} \\ $$$$\mathrm{with}\:{A},\:{B},\:{C}\:\mathrm{on}\:\mathrm{the}\:\mathrm{basic}\:\mathrm{circle} \\ $$
Commented by MJS last updated on 04/May/19
$$\mathrm{but}\:\mathrm{if}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{a}\:\mathrm{spheric}\:\mathrm{triangle}\:\mathrm{I}\:\mathrm{can}'\mathrm{t}\:\mathrm{help}... \\ $$
Commented by mr W last updated on 05/May/19
$${if}\:{spheric}\:{triangle}\:{is}\:{meant},\:{the}\:{max}. \\ $$$${spheric}\:{triangle}\:{ABC}\:{is}\:{when}\:{B}\:{and} \\ $$$${C}\:{are}\:{also}\:{on}\:{the}\:{base}\:{circle}\:\left({but}\:\Delta{ABC}\right. \\ $$$$\left.{must}\:{not}\:{be}\:{equilateral}\right).\:{the}\:{max}. \\ $$$${spheric}\:{triangle}\:{ABC}\:{is}\:{then}\:{equal}\:{to}\:{the} \\ $$$${total}\:{semisphere}\:{surface}. \\ $$
Answered by mr W last updated on 05/May/19
Commented by mr W last updated on 05/May/19
$${the}\:\Delta{ABC}\:{is}\:{maximum},\:{when}\:{its} \\ $$$${circumcircle}\:{is}\:{maximum}.\: \\ $$$${the}\:{maximum}\:{circumcircle}\:{is}\:{the} \\ $$$${base}\:{circle}\:{of}\:{the}\:{semisphere},\:{thus} \\ $$$${B}\:{and}\:{C}\:{should}\:{also}\:{lie}\:{on}\:{the}\:{rim}\:{of} \\ $$$${the}\:{semisphere}.\: \\ $$$${in}\:{the}\:{circumcircle}\:{the}\:{maximum} \\ $$$${triangle}\:{is}\:{an}\:{equilateral}\:{triangle}. \\ $$
Commented by ajfour last updated on 05/May/19
$$\mathrm{Thanks}\:\mathrm{for}\:\mathrm{the}\:\mathrm{light}\:\mathrm{again},\:\mathrm{Sirs}. \\ $$