Question 1204500
 



here we have a hyperbola 

{{{x^2/a^2-y^2/b^2=1}}}


with foci at:{{{M}}}({{{-500}}},{{{0}}}) =({{{-c}}},{{{0}}}) and {{{N}}}({{{500}}},{{{0}}})=({{{c}}},{{{0}}})


distance between foci is {{{2a}}}, so to get the distance multiply speed {{{400(m/(mu*s))}}} by a time {{{2(mu*s)}}}


{{{2a=2(mu*s)*400(m/(mu*s))=800m}}}

{{{a=400m}}}

since {{{c=500}}}, we have

{{{c^2=a^2+b^2}}}

{{{500^2=400^2+b^2}}}

{{{b^2=500^2-400^2}}}

{{{b^2=90000}}}

{{{b=300}}}


and your hyperbola is:


{{{x^2/400^2-y^2/300^2=1}}}

{{{x^2/160000 - y^2/90000 = 1}}}


What are the coordinates of the ship if it is {{{200m}}} from the shore({{{y=200}}})?

{{{x^2/160000 - 200^2/90000 = 1}}}

{{{x^2/160000 - 4/9 = 1}}}

{{{x^2/160000 = 1+4/9 }}}

{{{x^2/160000 = 13/9}}}

{{{x^2 = (13/9)160000}}}

{{{x^2 =2080000/9}}}

{{{x =sqrt(2080000/9)}}}

({{{x =400sqrt(13))/3}}} -> exactly

{{{x =480.74}}}-> approximately



 the coordinates of the ship if it is {{{200m }}}from the shore are ({{{480.74}}},{{{200}}})