Exponential Growth/Decay

Question

A bottle of seltzer water initially has a temperature of 79°F. It is left to cool in a refrigerator that has a

temperature of 38°F. After 10 minutes the temperature of the seltzer water is 66°F. Use​ Newton's Law of​ Cooling, T=C+(T0−C)e^kt​, to find a model for the temperature of the seltzer water​, ​T, after t minutes.

Answer 1

In order to solve this problem, first we must start with what we have as our givens:

Initial Temperature (T₀) = 79 deg F

Surrounding Temperature (C) = 38 deg F

Time (t) = 10 minutes

Temperature (T) after 10 minutes = 66 deg F

Based on Newton's Law of Cooling:

T(t) = C + (T₀ - C)e^kt

Looking at our given values, all that we now need to do is substitute them into the equation, which gives us:

66 deg F = 38 deg F + ( 79 deg F - 38 deg F)e^10k

So, if we solve this out we get:

28 deg F = ( 41deg F)e^10k

^--> 28 deg F / 41 deg F = e^10k

^--> 28 / 41 = e^10k

^--> ln (28 / 41) = ln (e^10k)

^--> ln (28 / 41) = 10k

k = (ln(28/41))/10

= - 0.03813675565

Which we would write as:

k = - 0.038

Answer 2

Plug in what you are given, and solve for k.