Question

After sitting on a shelf for a while, a can of soda at a room temperature (70, degrees70 ∘ F) is placed inside a refrigerator and slowly cools. The temperature of the refrigerator is 38, degrees38 ∘ F. Newton's Law of Cooling explains that the temperature of the can of soda will decrease proportionally to the difference between the temperature of the can of soda and the temperature of the refrigerator, as given by the formula below: T, equals, T, start subscript, a, end subscript, plus, left bracket, T, start subscript, 0, end subscript, minus, T, start subscript, a, end subscript, right bracket, e, start superscript, minus, k, t, end superscript T=T a ​ +(T 0 ​ −T a ​ )e −kt T, start subscript, a, end subscript, equalsT a ​ = the temperature surrounding the object T, start subscript, 0, end subscript, equalsT 0 ​ = the initial temperature of the object t, equalst= the time in minutes T, equalsT= the temperature of the object after tt minutes k, equalsk= decay constant The can of soda reaches the temperature of 62, degrees62 ∘ F after 15 minutes. Using this information, find the value of kk, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the can of soda, to the nearest degree, after 100 minutes. Enter only the final temperature into the input box.

Answer 1

The constant K is 0.019 and the temperature at 100 mins is 42.7⁰F.

How the temperature is determined.

Given that

`T = a + (T₀ - Ta) {e}^( - kt)`

T0 = intial temperature of can 70⁰F

Ta = temperature of the sorrounding = temperature of refrigerator = 38⁰F

T = 62⁰F, temperature of can after time t = 15mins

Substitute

`62 = 38 + (70 - 38) {e}^( - 15k)`

`24 = 32 {e}^( - 15k)`

`(24)/(32) = {e}^( - 15k)`

-15k = ln(24/32)

-15k = -0.2877

k = -0.2877/-15

k = 0.019

After 100 minutes

`T = 38 + (70 - 38) {e}^( - 0.019 * 100)`

T = 38 + 32* 0.1469

T = 38 + 4.7

T = 42.70⁰F