Question

A bottle of soda with a temperature of 71° Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of 35° F. After ten minutes, the internal temperature of the soda was 63° F. Use Newton’s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?

Answer 1

Consider the surrounding air temperature in the refrigerator is 71 degrees and the water temperature will decay exponentially towards 35 degree, that is;

T(t) = Aekt + 35

 

The initial temperature is 71. So that T(0) = 71 and substitute (0, 71) as follows:

        71 = Aek(0) + 35

71 – 35 = A

          A = 36

 

After one hour, that is 10 minutes the temperature is risen to 63 degrees, T(60) = 45,

 

Then;

        63 = 36ek(10) + 35

63 – 35 = 36ek(10)

   38/36 = ek(10)

   0.778 = ek(10)

 

Take natural log on both sides as follows:

ln(0.778) = ln{ek(10)}

ln(0.778) = k(10)

             k = ln(0.778)/10

               = -0.025

 

Thus the equation for cooling of water is T(t) = 36e(-0.025)t + 35.

 

Time required for the temperature after one hour is:

T(t) = 36e(-0.025)(60) + 35

      = 36e(-1.5) + 35

      = 43

 

Therefore, the temperature of soda after one year is 43 degrees.

Therefore, the temperature of soda after one year is 43 degrees.