Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Suppose $t$ is time, $T$ is the temperature of the object, and $T_s$ is the surrounding temperature. The following differential equation describes Newton's Law $$\frac{dT}{dt} = k (T-T_s),$$ where $k$ is a constant.

Suppose that we consider a $92^\circ\text{C}$ cup of coffee in a $20^\circ\text{C}$ room. Suppose it is known that the coffee cools at a rate of $1^\circ\text{C/min.}$ when it is $70^\circ\text{C}.$ Answer the following questions.

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1. Find the constant $k$ in the differential equation.
Answer (in per minute): $k =$

2. What is the limiting value of the temperature?
Answer (in Celsius): $T =$

3. Use Euler's method with step size $h=2$ minutes to estimate the temperature of the coffee after $10$ minutes.
Answer (in Celsius): $T(10) \approx$

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