Question

A cup of coffee which is initially at a temperature of 154∘F is placed in a room which is at a constant temperature of 71∘F. In 10 minutes, the coffee has cooled to 133∘F. Find the Newton's Law of Cooling formula that models the coffee's temperature in minutes. Keep at least 4 decimal places in your formula for rounded values. Determine to the nearest minute how long it will take the coffee to cool to a temperature of 100∘F.

Answer 1

Newton's law of cooling is
k (t₁ - t₂) = -ln (T₁ - T∞ / T₂ - T∞)

Use the two data points in the given to find k.

t = 0, T = 154
t = 10, T = 133

Solution:

k(0 - 10) = - ln (154 - 71/ 133 - 71)
-10k = -0.291716
k = 0.02917 or 0.0292

So now find t when T = 100
0.2917 * ( 0 - t) = -ln (154 - 71 / 100 -71)
- 0.02917t = - 1.0515
t = 36. 05 minutes
to the nearest minute t = 36