Answer :
Final Answer:
According to the given scenario, time taken to reach coffee is:
a) At 175 degrees, approximately 36 minutes
b) At 170 degrees, approximately 58 minutes
c) At 165 degrees, approximately 88 minutes
d) 160 degrees, approximately 131 minutes
Explanation:
The rate at which the coffee cools down follows Newton's law of cooling, which states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings. As the temperature difference decreases, the rate of cooling also decreases.
In this scenario, we can use the formula for Newton's law of cooling to calculate the time it takes for the coffee to reach certain temperatures.
To calculate the time it takes for the coffee to reach a specific temperature, we need to find the rate constant (k) from the given information and then use it in the formula. The rate constant can be found by using the formula:
[tex]\( k = \frac{ln\left(\frac{T_{\text{final}} - T_{\text{room}}}{T_{\text{initial}} - T_{\text{room}}}\right)}{t} \)[/tex]
Where [tex]\( T_{\text{final}} \)[/tex] is the final temperature, [tex]\( T_{\text{initial}} \)[/tex] is the initial temperature, [tex]\( T_{\text{room}} \)[/tex] is the room temperature, and t is the time in minutes.
Using this formula, we can find the rate constant and then use it to find the time it takes for the coffee to reach the desired temperatures. For each temperature, plug in the values and solve for t. These calculations yield the approximate times required for the coffee to cool to each specified temperature.
