Final answer:
To find the temperature of the coffee after 70 minutes, we can use Newton's Law of Cooling, and the same principle can be applied to find the time it takes for the coffee to cool to 120°F.
Step-by-step explanation:
To solve this problem, we can use Newton's Law of Cooling, which states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature.
a. After half an hour, the temperature of the coffee is 170° F. We can use this information to set up the equation:
T - 70 = (190 - 70) * e^(-k * 0.5)
where T is the unknown temperature, k is the rate constant, and e is Euler's number. Solving for T, we find that T = 170 + 20 * e^(-k * 0.5). Now we can plug in 70 minutes for the time and solve for T:
T = 170 + 20 * e^(-k * 70/60)
b. To find the time it takes for the coffee to cool to 120° F, we set up the equation:
120 - 70 = (190 - 70) * e^(-k * t)
where t is the unknown time. Solving for t, we find that t = ln(5/12) / (-k). Now we can plug in 120° F for the temperature and solve for t:
t = ln(5/12) / (-k)