High School

You place a cup of hot chocolate at 220 degrees F on a table in a room that is 72 degrees F. Twelve minutes later, it is 205 degrees F. Approximately how long will it be before the hot chocolate reaches 175 degrees F? Round your answer to the nearest minute.

Use Newton's Law of Cooling: [tex]T(t) = T_a + (T_0 - T_a)e^{-kt}[/tex].

Answer :

T(t)= 205 F
T o = 220 F
T a = 72 F
t = 12 min;
Here: k ( constant ) = ?
205 = 72 + ( 220 - 72 ) * e ^( - 12 k )
133 = 148 * e ^(-12 k )
e^(-12k) = 0.89865
- 12 k = ln 0.86865
- 12 k = - 0.1068
k = 0.0089
Then we will plug in the formula:
175 = 72 + ( 220 - 72 ) * e^(-0.0089 t )
103 = 148 * e^(-0.0089 t )
e^(-0.0089 t ) = 0.6959
- 0.0089 t = ln 0.6959
- 0.0089 t = - 0.36255
t = 0.369255 : 0.0089
t = 40.73 ≈ 41 min
Answer: It will be 41 minutes later.

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