Question

A cooler has a temperature of 32 degrees Fahrenheit. A bottled drink is placed in the cooler with an initial temperature of

70 degrees Fahrenheit. The function

`f(t) = {ce}^( (- kt)) + 32`
,represents the situation, where t is time in minutes, C is a
constant, and k is a constant.
After 3 minutes the bottle has a temperature of 42 degrees. What is the approximate value of k?​

Answer 1

`k \approx 0.44`

Explanation:

Given function:

`f(t) = (ce)^(-kt)+32`

As per question statement:

Initial temperature of bottle is 70
`^\circ F`.

i.e. when time = 0 minutes, f(t) = 70
`^\circ F`

`70 = ce^(-k* 0)+32\\\Rightarrow 38 = ce^(0)\\\Rightarrow c = 38`

After t = 3, f(t) = 42
`^\circ F`

`42 = 38 * e^(-k* 3)+32\\\Rightarrow 42-32 = 38 * e^(-3k) \\\Rightarrow 10 = 38 * e^(-3k) \\\Rightarrow e^(3k) = (38)/(10)\\\Rightarrow e^(3k) = 3.8\\\\\text{Taking } log_e \text{both the sides:}\\\\\Rightarrow log_e {e^3k} = log_e {3.8}\\\Rightarrow 3k * log_ee=log_e {3.8} (\because log_pq^r=r * log_pq)\\\Rightarrow 3k * 1=log_e {3.8}\\\Rightarrow 3k = 1.34\\\Rightarrow k = (1.34)/(3)\\\Rightarrow k \approx 0.44`

Hence, the value is:

`k \approx 0.44`