Precalculus question

Question

A can of soda at 39 °F is removed from a refrigerator and placed in a room where the air temperature is 68 °F. After 13 minutes, the temperature of the can has risen to 49 °F. How many minutes after the can is removed from the refrigerator will its temperature reach 60 °F? 

Round your answer to the nearest whole minute.

Answer 1

Newton's Law of Cooling/Heating states T = T_a+(T₀-T_a)e^-kt where T_a=68°F is the air temperature and T₀=39°F is the initial temperature of the object. Given that the temperature of the turkey rose to T=49°F after t=13 minutes, then we can solve for k and use that value to answer how long it'll take it to rise to T=60°F:

T = T_a+(T₀-T_a)e^-kt

49 = 68+(39-68)e^-k(13)

-19 = -29e^-13k

19/29 = e^-13k

ln(19/29) = -13k

(-1/13)ln(19/29) = k

k ≈ 0.0325

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

T = T_a+(T₀-T_a)e^-kt

60 = 68+(39-68)e^{-(-1/13)ln(19/29)t}

-8 = -29e^{(1/13)ln(19/29)t}

8/29 = e^{(1/13)ln(19/29)t}

ln(8/29) = (1/13)ln(19/29)t

ln(8/29)/[(1/13)ln(19/29)] = t

t ≈ 39.59 ≈ 40

Thus, it will take approximately 40 minutes for the can of soda to rise in temperature from 39°F to 60°F.

Hope the answer and explanations helped!