Differential equation word problem I'm doing a set of computer science programming questions and this was one, though I think it's simply solvable with differential equations, but I'm not too good with them. Any help?

Mr. Chickenorfish prepares a cup of coffee, then promptly forgets about it. Initially, the coffee is at $210$ F. Fifteen minutes later, the coffee is at $190$ F. If room temperature is $72$ F, when will the coffee be at $150$ F?

 A: You need a model for how the temperature decreases.  I will guess that the heat loss is proportional to the temperature difference between the coffee and room temperature.  Let the coffee temperature be $T$.  My model says $\frac {dT}{dt}=k(T-72)$ and $T(0)=210, T(15)=190$.  You need to solve the differential equation, use the two pieces of data to evaluate $k$ and the integration constant, then find the time when $T=150$
A: It is just exponential decay. The temperature will be $\tau(t) = 72+(210-72)e^{-\lambda t}$. You are given $\tau(15) = 190$ which lets you compute $\lambda$, then you are asked to solve $\tau(t) = 150$ for $t$.
Solve for $\lambda$:

$\tau(15) = 190$ gives$\frac{190-72}{210-72} = e^{-\lambda 15}$, or equivalently $\ln(\frac{190-72}{210-72}) = -15\lambda$.

Solve for $t$:

Then solve for $t$ in $\frac{150-72}{210-72} = e^{-\lambda t}$, or equivalently $\ln(\frac{150-72}{210-72}) = -\lambda t$.

(Or, you could avoid computing $\lambda$ and just divide the above to get an expression for $\frac{t}{15}$.)
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