You place a cup of 220 degrees f hot chocolate on a table in a room that is 72 degrees f, and 12 minutes later, it is 205 degrees f. approximately how long will it be before the coffee is 175 degrees f? round to the nearest minute. (use newtons law of cooling: t(t)=ta+(t0-ta)e^-kt

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
It will be 41 minutes later.

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