Suppose that a cup of coffee begins at 178 degrees and, after sitting in room temperature of 69 degrees for 16 minutes, the coffee reaches 172 degrees. Using Newton's Law of Cooling, how long will it take before the coffee reaches 161 degrees?
Include at least 2 decimal places in your answer.
minutes

Step 1 ：Given that a cup of coffee starts at 178 degrees and cools to 172 degrees in 16 minutes in a room with a temperature of 69 degrees, we can use Newton's Law of Cooling to calculate the cooling constant, k. Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature. The formula for Newton's Law of Cooling is: \(\frac{dT}{dt} = -k(T - T_a)\), where \(\frac{dT}{dt}\) is the rate of change of the temperature, T is the temperature of the object, \(T_a\) is the ambient temperature, and k is the cooling constant.

Step 2 ：We can rearrange this formula to solve for k: \(k = -\frac{dT}{dt} / (T - T_a)\). Substituting the given values into the formula, we find that \(k = 0.0035386808749692457\).

Step 3 ：Next, we can use the value of k to calculate how long it will take for the coffee to cool to 161 degrees. We can do this by rearranging the formula to solve for t: \(t = -\frac{1}{k} \ln\left(\frac{T - T_a}{T_0 - T_a}\right)\), where t is the time, T is the final temperature (161 degrees), \(T_0\) is the initial temperature (178 degrees), and ln is the natural logarithm.

Step 4 ：Substituting the given values into the formula, we find that \(t = 47.91596393432255\) minutes.

Step 5 ：Rounding to two decimal places, we find that it will take approximately \(\boxed{47.92}\) minutes for the coffee to cool to 161 degrees.