Answer: The temperature of the liquid after 7 hours will be 84.7 °F
Solution:
Initial Temperature of the liquid is 170°F:
at t=0, T(0)=170°F
Temperature of the air: Ta=76°F
k=0.34
Newton's cooling model:
T(t)=Ta+C e^(-kt)
Replacing the known values:
T(t)=76+C e^(-0.34t)
We know at t=0, T(0)=170
t=0→T(0)=76+C e^(-0.34(0))
170=76+C e^0
170=76+C (1)
170=76+C
Solving for C: Subtracting 76 both sides of the equation:
170-76=76+C-76
94=C
C=94
Then, the Newton's cooling model is:
T(t)=76+94 e^(-0.34t)
The temperature of the liquid after 7 hours will be:
t=7→T(7)=76+94 e^(-0.34(7))
T(7)=76+94 e^(-2.38)
T(7)=76+94 (0.092550578)
T(7)=76+8.699754332
T(7)=84.69975433
T(7)=84.7
Answer: The temperature of the liquid after 7 hours will be 84.7 °F