Suppose you have just poured a cup of freshly brewed coffee with temperature $\displaystyle {90}^{{\circ}}{C}$ in a room where the temperature is $\displaystyle {20}\circ{C}$.
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, $\displaystyle {T}{\left({t}\right)}$, satisfies the differential equation
$\displaystyle {\frac{{{d}{T}}}{{{\left.{d}{t}\right.}}}}={k}{\left({T}-{T}_{{\text{room}}}\right)}$
where $\displaystyle {T}_{{\text{room}}}={20}\circ{C}$ is the room temperature, and $\displaystyle {k}$ is some constant.
Suppose it is known that the coffee cools at a rate of $\displaystyle {2}\circ{C}$ per minute when its temperature is $\displaystyle {70}\circ{C}$.

A. What is the limiting value of the temperature of the coffee?
$\displaystyle \lim_{{{t}\to\infty}}{T}{\left({t}\right)}=$ Preview Question 6 Part 1 of 4  

B. What is the limiting value of the rate of cooling?
$\displaystyle \lim_{{{t}\to\infty}}{\frac{{{d}{T}}}{{{\left.{d}{t}\right.}}}}=$ Preview Question 6 Part 2 of 4  

C. Find the constant $\displaystyle {k}$ in the differential equation.
$\displaystyle {k}=$ Preview Question 6 Part 3 of 4   .

D. Use Euler's method with step size $\displaystyle {h}={3}$ minutes to estimate the temperature of the coffee after $\displaystyle {15}$ minutes.
$\displaystyle {T}{\left({15}\right)}=$ Preview Question 6 Part 4 of 4   .