Question 1178344
Newton's law of cooling states that 

{{{dT/dt = -k(T-T[a])}}},


where T is the temperature of the object and {{{T[a]}}} is the temperature of the surrounding.

==>{{{dT/(T-75) = -k*dt}}}, or {{{ln(T-75) = -kt + c}}} after integrating both sides wrt t.

Now T(0) = 185 ==> ln(185 - 75) = -k*0 + c  <==> c = ln(110) ==>  {{{ln((T-75)/110) = -kt}}} 


T(30) = 150 ==> {{{ln((150-75)/110) = -30k}}}, or {{{(-1/30)*ln(15/22) = k}}}

==> {{{k  = (1/30)*ln(22/15)}}}

==> {{{ln((T-75)/110) = -(1/30)*ln(22/15)*t}}} , or

==> {{{(T-75)/110 = e^(-(1/30)*ln(22/15)*t)}}}

After t = 50 minutes, {{{(T-75)/110 = e^(-(50/30)*ln(22/15))}}}

==> {{{highlight(T = 133.1^0)}}}F

The roast turkey will cool to 105 degrees F after 

{{{ln((105-75)/110) = -(1/30)*ln(22/15)*t}}}, or {{{highlight(t = -(30*ln(3/11))/ln(22/15) = 101.77)}}} minutes.