Question 741083
The 1st plane has a head start of 
{{{ d[1] = 200*(1/2) }}}
{{{ d[1] = 100 }}} mi
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Start a stopwatch when the 2nd plane leaves
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Let {{{ t }}} = time on stopwatch when they
are {{{ 300 }}} miles apart
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distance for 1st plane:
{{{ 100 + 200t }}}
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Distance for 2nd plane:
{{{ 150t }}}
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The planes travel 90 degrees to each other, so
{{{ ( 100 + 200t )^2 + (150t)^2 = 300^2 }}}
{{{ 100^2*( 1 + 2t )^2 + 22500t^2 = 90000 }}}
{{{ 10000*( 1 + 4t + 4t^2) + 22500t^2 = 90000 }}}
{{{ 10000 + 40000t + 40000t^2 + 22500t^2 = 90000 }}}
{{{ 62500t^2 + 40000t - 80000 = 0 }}}
{{{ 625t^2 + 400t - 800 = 0 }}}
{{{ 25t^2 + 16t - 32 }}}
Use quadratic formula
{{{ t = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}} 
{{{ a = 25 }}}
{{{ b = 16 }}}
{{{ c = -32 }}}
{{{ t = (-16 +- sqrt( 16^2 - 4*25*(-32) )) / (2*25) }}} 
{{{ t = (-16 +- sqrt( 256 + 3200 )) / 50 }}} 
{{{ t = (-16 +- sqrt( 3456 )) / 50 }}} 
{{{ t = ( -16 + 58.788 ) / 50 }}}
{{{ t = 42.788 / 50 }}}
{{{ t = .856 }}} hrs
{{{ 6.5 + .856 = 7.356 }}}
{{{ .356*60 = 22 }}} min
They will be out of range at 7:22
check:
1st plane:
{{{ d = 100 + 200*.856 }}}
{{{ d = 100 + 171.2 }}}
{{{ d = 271.2 }}}
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2nd plane:
{{{ d = 150*.856 }}}
{{{ d = 128.4 }}}
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{{{ 271.2^2 + 128.4^2 = 300^2 }}}
{{{ 73549.4 + 16486.6 = 90000 }}}
{{{ 90036 = 90000 }}}
Maybe error is due to rounding off.
Recheck my math- I think the method
is OK