Math help, Exponential and Logarithmic Models

Question

A roasted turkey is taken from an oven when its temperature has reached 185∘185∘ Fahrenheit and is placed on a table in a room where the temperature is 75∘75∘ Fahrenheit. Give answers accurate to at least 2 decimal places.

(a) If the temperature of the turkey is 154∘154∘ Fahrenheit after half an hour, what is its temperature after 45 minutes?

 Fahrenheit

(b) When will the turkey cool to 100∘100∘ Fahrenheit?

 hours.

Answer 1

Newton's Law of Cooling states T = T_a+(T₀-T_a)e^-kt where T_a=75°F is the air temperature and T₀=185°F is the initial temperature of the object. Given that the temperature of the turkey cooled to T=154°F after t=30 minutes, then we can solve for k and use that value to answer how long it'll take it to cool down to T=100°F:

T = T_a+(T₀-T_a)e^-kt

154 = 75+(185-75)e^-k(30)

79 = 110e^-30k

79/110 = e^-30k

ln(79/110) = -10k

(-1/10)ln(79/110) = k

k ≈ 0.0331

Now that we have k, we can then solve for t:

T = T_a+(T₀-T_a)e^-kt

100 = 75+(185-75)e^{-(-1/10)ln(79/110)t}

25 = 110e^{(1/10)ln(79/110)t}

25/110 = e^{(1/10)ln(79/110)t}

ln(25/110) = (1/10)ln(79/110)t

ln(7/75)/[(1/10)ln(79/110)] = t

t ≈ 71.642

Thus, it will take approximately 71.642 minutes for the turkey to cool from 154°F to 100°F.

To figure out what temperature the turkey will cool down to after t=45 minutes, we compute the following:

T = 75+(185-75)e^{-(-1/10)ln(79/110)(45)} ≈ 99.8°F

Hope the answer and explanations helped!