A freshly brewed cup of coffee has temperature 95°C in a 20°C room. When its temperature is 68°C, it is cooling at a rate of 1°C per minute. When does this occur? (Round your answer to two decimal places.)
To solve this problem, we can use the concept of thermal equilibrium, which states that the rate of heat lost by an object is equal to the rate of heat gained by its surroundings.
Let's break down the information given in the question:
- The initial temperature of the coffee is 95°C.
- The temperature of the room is 20°C.
- The rate at which the coffee is cooling is 1°C per minute.
Using the formula for thermal equilibrium, we can calculate how long it takes for the coffee to reach a temperature of 68°C.
The rate of heat lost by the coffee is determined by the temperature difference between the coffee and the room:
Rate of heat lost = k * (Temperature of coffee - Temperature of room)
The rate of heat gained by the room is given by:
Rate of heat gained = k * (Temperature of room - Temperature of coffee)
Since the rates of heat lost and gained are equal at thermal equilibrium, we can set them equal to each other:
k * (Temperature of coffee - Temperature of room) = k * (Temperature of room - Temperature of coffee)
Simplifying the equation:
(Temperature of coffee - Temperature of room) = (Temperature of room - Temperature of coffee)
2 * Temperature of coffee = 2 * Temperature of room
Temperature of coffee = Temperature of room
So, at thermal equilibrium, the temperature of the coffee will be equal to the temperature of the room.
Therefore, the coffee will reach a temperature of 68°C when the room also reaches a temperature of 68°C.
Since the rate of cooling is 1°C per minute, and the initial difference between the coffee and the room is 75°C (95°C - 20°C), it will take 75 minutes for the coffee to cool down from 95°C to 20°C. During this time, the temperature of the coffee will decrease by 1°C per minute.
Therefore, the coffee will reach a temperature of 68°C after 68 minutes.
Therefore, the answer is 68 minutes.