A hot cup of coffee is 200F when poured and placed in a room that is 70F. 10 min later the temperature of the coffee has cooled to 180F. What is the temperature of the coffee 10…

Question

asked Feb 19, 2020 55.2k views

2 Answers

Lemieuxster · Answer 1 · 2020-02-24T03:53:24+0000

Answer:

The temperature of coffee becomes 163.08 Fahrenheit.

Explanation:

The Newton's law of cooling can be mathematically written as

$-(dT)/(dt)=k(T-T_(room))\\\\(-dT)/((T-T_(room)))=kdt$

Applying the limits in the above relation as

At t = 0 ; T= 200 F

At t =10 ; T = 180 F

$-\int_(200)^(180)(dT)/(T-70)=\int_(0)^(10)kdt\\\\-ln(T-70)_(200)^(180)=k* 10\\\\\therefore k=(ln(1.181))/(10)\\\\\therefore k=0.0167$

Applying the limits in the above relation as

At t = 10 ; T= 180 F

At t =20 ; T = T F

$-\int_(180)^(T)(dT)/(T-70)=\int_(10)^(20)0.0167dt\\\\-ln(T-70)_(180)^(T)=0.167\\\\\therefore T = 163.08F$