Question 1112016
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A roasted turkey is taken from an oven when its temperature has reached 185 Fahrenheit and is placed on a
table in a room where the temperature is 75 Fahrenheit.

a). If the temperature of the turkey is 141 Fahrenheit after half an hour, what is its temperature after 45 minutes?

b). When will the turkey cool to 100 Fahrenheit?
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You can use Newton's Law of Cooling.......any of the following 3 formulae should work: {{{highlight_green(system(highlight(matrix(1,3, "T(t)", "=", T[s] + (T[o] - T[s]) * e^(- kt))), OR, highlight(matrix(1,3, dT/dt, "=", - k(T[i] - T[o]))), OR, highlight(matrix(1,3, k(t[1] - t[2]), "=", - ln((T[1] - T[infinity])/(T[2] - T[infinity])))))))}}}

This author's preference is the 1<sup>st</sup> formula. In this case, the cooling rate (k) is first needed, and is derived as follows:

       {{{highlight(highlight_green(highlight(matrix(1,3, "T(t)", "=", T[s] + (T[o] - T[s]) * e^(- kt)) )))}}}, where: {{{t}}} = time taken to get to a COOLED temperature ({{{1/2}}} an hr, or 30 minutes, in this case)
                                             {{{T(t)}}} = TEMPERATURE (T) at a given time (t)___(141<sup>o</sup>F, in this case) 
                                              {{{T[s]}}} = SURROUNDING Temperature (75<sup>o</sup>F, in this case)
                                              {{{T[o]}}} = ORIGINAL/INITIAL temperature (185<sup>o</sup>F, in this case)
                                               {{{k}}} = the CONSTANT or COOLING rate (UNKNOWN, in this case)
                                      {{{highlight(highlight_green(highlight(matrix(1,3, "T(t)", "=", T[s] + (T[o] - T[s]) * e^(- kt)) )))}}}
                                      {{{matrix(1,3, "T(30)", "=", 75 + (185 - 75) * e^(- k*30))}}} ---- Substituting 30 for t, 75<sup>o</sup> for {{{T[s]}}}, and 185<sup>o</sup> for {{{T[o]}}}
                                          {{{matrix(1,3, 141, "=", 75 + (185 - 75) * e^(- 30k))}}} ----- Substituting 141<sup>o</sup> for T(30)
                                          {{{matrix(4,3, 141, "=", 75 + 110e^(- 30k), 66, "=", 110e^(- 30k), 66/110, "=", e^(- 30k), 3/5, "=", e^(- 30k))}}}
                                       {{{matrix(1,3, - 30k, "=", ln (3/5))}}} ----- Converting to NATURAL LOGARITHMIC (ln) form
                                              {{{matrix(1,5, k, "=", ln(3/5)/(- 30), "=", .01702752)}}}
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a). If the temperature of the turkey is 141 Fahrenheit after half an hour, what is its temperature after 45 minutes?

                                                           {{{highlight(highlight_green(highlight(matrix(1,3, "T(t)", "=", T[s] + (T[o] - T[s]) * e^(- kt)) )))}}}
                                                           {{{matrix(1,3, "T(45)", "=", 75 + (185 - 75) * e^(- .01702752*45))}}} ---- Substituting 45 for t, 75<sup>o</sup> for {{{T[s]}}}, 185<sup>o</sup> for {{{T[o]}}},          
                                                                                                                                               and .01702752 for k
Temperature, after 45 minutes, or {{{matrix(1,5, "T(45)", "=", 75 + 110e^(- .01702752*45),
"=", 126.1233802)}}}, or approximately 126<sup>o</sup>F.
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b). When will the turkey cool to 100 Fahrenheit?

                                                           {{{highlight(highlight_green(highlight(matrix(1,3, "T(t)", "=", T[s] + (T[o] - T[s]) * e^(- kt)) )))}}}
                                                                {{{matrix(1,3, 100, "=", 75 + (185 - 75) * e^(- .01702752t))}}} ---- Substituting 100 for T(t), 75<sup>o</sup> for {{{T[s]}}}, 185<sup>o</sup> for {{{T[o]}}},
                                                                                                                                            and .01702752 for k
                                                        {{{matrix(5,3, 100, "=", 75 + 110e^(- .01702752t), 100 - 75, "=", 110e^(- .01702752t), 25, "=", 110e^(- .01702752t),
25/110, "=", e^(- .01702752t),
5/22, "=", e^(- .01702752t))}}}
                                             {{{matrix(1,3, - .01702752t, "=", ln(5/22))}}} ---- Converting to NATURAL LOGARITHMIC (ln) form
Time taken for the turkey to cool to 100<sup>o</sup>F, or {{{matrix(1,11, t, "=", ln(5/22)/(- .01702752),
"=", "87.0123466,", or, approximately, 87, "minutes,", 7.5, seconds)}}}</font></font></font></b></pre>