Suppose you have just poured a cup of freshly brewed coffee with temperature $90^{\circ} C$ in a room where the temperature is $25^{\circ} C$.
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. Therefore, the temperature of the coffee, $T(t)$, satisfies the differential equation $$\frac{dT}{dt} = k(T-T_{\rm room})$$ where $T_{\rm room}=25$ is the room temperature, and $k$ is some constant.
Suppose it is known that the coffee cools at a rate of $1^{\circ} C$ per minute when its temperature is $75^{\circ} C$.

A. What is the limiting value of the temperature of the coffee?
$\displaystyle \lim_{t\to\infty} T(t) =$

B. What is the limiting value of the rate of cooling?
$\displaystyle \lim_{t\to\infty} \frac{dT}{dt} =$

C. Find the constant $k$ in the differential equation.
$k=$ .

D. Use Euler's method with step size $h=3$ minutes to estimate the temperature of the coffee after $15$ minutes.
$T(15)=$ .