Question

this exercise uses newtons law of cooling.
a roasted turkey is taken from an oven when its temperature has reached 185°f and is placed on a table in a room where the temperature is 72°f.
find the following values for newtons law of cooling (t(t) = t_s + d_0e^{-kt}), where time is measured in minutes and the temperature is measured in degrees fahrenheit, if the temperature of the turkey is 145°f after half an hour.
t_s = 72
d_0 = 113
k=
(a) what is its temperature (in °f) after 45 min? (round your answer to the nearest whole number.)
(b) after how many hours will the turkey cool to 100°f? (round your answer to one decimal place.)

Explanation:

Step1: Find the value of \(k\)

We know that \(T(t)=T_{s}+D_{0}e^{-kt}\). When \(t = 30\) (half - an hour), \(T(30)=145\), \(T_{s}=72\), \(D_{0}=113\).
Substitute into the formula: \(145 = 72+113e^{-30k}\).
First, subtract \(72\) from both sides: \(145 - 72=113e^{-30k}\), so \(73 = 113e^{-30k}\).
Then, divide both sides by \(113\): \(\frac{73}{113}=e^{-30k}\).
Take the natural logarithm of both sides: \(\ln(\frac{73}{113})=- 30k\).
Solve for \(k\): \(k=-\frac{1}{30}\ln(\frac{73}{113})\approx-\frac{1}{30}\times(-0.457)\approx0.0152\).

Step2: Find the temperature after \(t = 45\) minutes

We have \(T(t)=72 + 113e^{-0.0152t}\).
Substitute \(t = 45\) into the formula: \(T(45)=72+113e^{-0.0152\times45}\).
First, calculate \(-0.0152\times45=-0.684\).
Then, \(e^{-0.684}\approx0.505\).
So \(T(45)=72 + 113\times0.505=72+57.065\approx129\).

Step3: Find the time \(t\) when \(T(t)=100\)

Set \(T(t)=100\), so \(100=72 + 113e^{-0.0152t}\).
Subtract \(72\) from both sides: \(100 - 72=113e^{-0.0152t}\), \(28 = 113e^{-0.0152t}\).
Divide both sides by \(113\): \(\frac{28}{113}=e^{-0.0152t}\).
Take the natural logarithm of both sides: \(\ln(\frac{28}{113})=-0.0152t\).
Solve for \(t\): \(t=-\frac{1}{0.0152}\ln(\frac{28}{113})\).
Since \(\ln(\frac{28}{113})\approx-1.45\), \(t=-\frac{1}{0.0152}\times(-1.45)\approx95.4\).

Answer:

\(k\approx0.015\)
(a) \(129^{\circ}F\)
(b) \(t\approx95.4\) minutes