After heating up in a teapot, a cup of hot water is poured at a temperature of 203^\circ203 ∘
F. The cup sits to cool in a room at a temperature of 72^\circ72
∘
F. Newton's Law of Cooling explains that the temperature of the cup of water will decrease proportionally to the difference between the temperature of the water and the temperature of the room, as given by the formula below:
T=T_a+(T_0-T_a)e^{-kt}
T=T
a

+(T
0

−T
a

)e
−kt

T_a=T
a

= the temperature surrounding the object
T_0=T
0

= the initial temperature of the object
t=t= the time in minutes
T=T= the temperature of the object after tt minutes
k=k= decay constant

The cup of water reaches the temperature of 183^\circ183
∘
F after 2.5 minutes. Using this information, find the value of kk, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the cup of water, to the nearest degree, after 5 minutes.