Difference T - C, where C is the constant temperature of the surrounding medium?
That is: dT/dt= -k(T-C)
i) dT/dt = d(a(e^(-kt))+C)/dt= -ake^(-kt) = -k(ae^(-kt)).
T= a(e^(-kt))+C, so T-C = ae^(-kt)
This implies that dT/dt = -k(ae^(-kt)) = -k(T-C).
So T= T(t) satisfies the equation.
ii) lim (t-> inf) T(t) = C. This represents the room temperature, the temperature which the coffee approaches over time. So C = 72.
When t = 0, the problem tells us that T = 130 (Freshly brewed => t=0.)
When t= 4.3, T = 120.
So T(0) = 130, T(4.3) = 120. We use these two pieces of info to determine t such that T(t) = 105.
T(0) = a +72 = 130. This implies that a = 58.
We now have T(t) = 58e^(-kt)+72.
T(4.3) = 58e^(-4.3k)+72 = 120. This implies that k is about 0.044
We now have T(t) = 58e^(-.044t)+72.
Finally we wish to solve T(t) = 105 for t.
58e^(-.044t)+72 = 105 => t = 12.8167.
So your coffee will be ready in about 12.8 minutes