Answer:
16.033 min
Step-by-step explanation:
We can write Newton's Law of Cooling as
T = Tₐ + (T₀ - Tₐ)e^(-kt), where
Tₐ = ambient temperature of the surroundings
T₀ = initial temperature of the object
k = the cooling constant
t = the time
(1) Calculate the cooling constant
Data:
T = 110 °F
T₀ = 180 °F
Tₐ = 70 °F
t = 20 min
Calculation:
110 = 70 + (180 - 70)e^(-k×20)
40 = 110e^(-20k)
0.363 64 = e^(-20k)
-1.0116 = -20k
k = 0.050 580 min⁻¹
(2)After adding cream
(a) Calculate T₀
2 oz × 40 °F + 12 oz × 180 °F = 14 oz × T₀
80 °F + 2160°F = 14T₀
2240 °F = 14T₀
T₀ = 160 °F
(b) Calculate the time to cool to drinking temperature
110 = 70 + (160 - 70)e^(-0.050 580t)
40 = 90e^(-0.050 580t)
0.444 44 = e^(-0.050 580t)
-0.810 93 = -0.050580t
t = 16.033 min
It will take 16.033 min for the mixture to reach drinking temperature
