Answer:
27 mins
Step-by-step explanation:
Assuming the relationship between Temperature and time follows Newton's law of cooling:
T (t) = T₀ + T₁ exp (rt)-------------------------------------------------------------- (1)
Where T₀ = Original temperature of the room (°F) ; 72°F
T₁ = Temperature of the roast (°F); 165-72 =93°F
Please note that we have to subtract the room to get the absolute temperature of the roast.
r = Rate of cooling (°F/min)
t = time (min)
T (t) = 72 + 93 exp (rt)-------------------------------------------------------------- (2)
After 10 minutes, the temperature of the roast drops to 145°F. Here t =10 mins and T(10) = 145 °F
Substituting into (2) we have :
T(10) = 72 +93 exp(10r)
145 = 72 + 93 exp(10r)
(145-72)/93 = exp (10r)
Taking the natural logarithm of both sides, we have:
In [(145-72)/93 ] = 10r
-0.24214 =10r
r = -0.02421 °F/min
Substituting into equation (2), the complete equation as a function of time becomes:
T (t) = 72 + 93 exp (-0.024214t)--------------------------------------------- (3)
To find the time taken to reach 120°F, we substitute into equation (3)
120 = 72 +93 exp (-0.024214t)
(120-72)/93= exp (-0.024214t)
In[(120-72)/93] =-0.024214t
-0.66139 = -0.024214t
t = -0.66139/ -0.024214
=27.3143
≈ 27 mins
