Final answer:
To find the temperature of the coffee 15 minutes after being made, we can use Newton's Law of Cooling. The initial temperature of the coffee can be calculated using the temperature of the coffee after 20 minutes and the room temperature. Then, we can substitute the value of k and solve for the temperature of the coffee 15 minutes after being made.
Step-by-step explanation:
In this problem, we can use Newton's Law of Cooling to determine the temperature of the coffee 15 minutes after being made. The law states that the rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding temperature.
We know that after 20 minutes, the temperature of the coffee was 68°C, and the room temperature is given as 20°C. We can set up the equation:
(Temperature of the coffee after 20 minutes) - (Room temperature) = (Initial temperature of the coffee) - (Room temperature)e^(-kt)
Substituting the known values into the equation:
68 - 20 = (Initial temperature of the coffee) - 20e^(-k*20)
Simplifying the equation leads to (Initial temperature of the coffee) = 68 - 20e^(-k*20) + 20
Next, we need to find the value of k. To do this, we can use the information that the temperature of the coffee after 15 minutes was 75°C. Substituting this value into the equation and solving for k gives us k ≈ 0.1235.
Finally, we can substitute the value of k and solve for the temperature of the coffee 15 minutes after being made:
(Temperature of the coffee after 15 minutes) - 20 = (Initial temperature of the coffee) - 20e^(-0.1235*15)
Simplifying the equation gives us (Temperature of the coffee after 15 minutes) = 75 - 20e^(-0.1235*15) + 20.