Calculating Outside Temperature Using Exponential Decay Model

[Math10]

Homework Statement

A cup of boiling water is placed outside at 1:00 PM. One minute later the temperature of the water is 152 degrees Fahrenheit. After another minute its temperature is 112 degrees Fahrenheit. Find the outside temperature.

Homework Equations

None.

The Attempt at a Solution

T(t)=Ta+(To-Ta)e^(-kt)
T(t)=Ta+(212-Ta)e^(-kt)
T(1)=152=Ta+(212-Ta)e^(-k)
T(2)=112=Ta+(212-Ta)e^(-2k)
152-Ta=(212-Ta)e^(-k)
112-Ta=(212-Ta)e^(-2k)
112/152=e^(-k)
e^k=152/112
k=ln(19/14)
Now I'm stuck. What do I do?

 

[ehild]

Homework Statement

A cup of boiling water is placed outside at 1:00 PM. One minute later the temperature of the water is 152 degrees Fahrenheit. After another minute its temperature is 112 degrees Fahrenheit. Find the outside temperature.

The Attempt at a Solution

T(t)=Ta+(To-Ta)e^(-kt)
T(t)=Ta+(212-Ta)e^(-kt)
T(1)=152=Ta+(212-Ta)e^(-k)
T(2)=112=Ta+(212-Ta)e^(-2k)
152-Ta=(212-Ta)e^(-k)
112-Ta=(212-Ta)e^(-2k)
112/152=e^(-k)

That is wrong. Correctly: $\frac{112-T_a}{152-T_a}=e^{-k}$
Substituting the exponent into the first equations eliminates the unknown k:

$152-Ta=(212-Ta)\frac{112-T_a}{152-T_a}$

Solve for Ta.

 

[Math10]

Thank you so much for the help! I got the right answer!