A pan of cold water with a temeprature of 35 degrees F is brought into a room with a temperature of 75 degrees F. After one hour, the temperature in the pan is 45 degrees F. a) Write a formula for the temperature (T) as a function of time (according to Newton's Law of Cooling). b) Find the temperature of the water after 3 hours. c) how long will it take the water in the pan to reach 60 degrees F, counting since the moment it was brought into the room.
a) According to Newton's Law of Cooling, the rate of change of temperature of an object is proportional to the difference between its current temperature and the surrounding temperature.
We can express this as:
dT/dt = -k(T - T_s)
where:
- dT/dt is the rate of change of temperature (how fast it's changing)
- T is the temperature of the object at time t
- T_s is the surrounding temperature
- k is the cooling constant (a positive constant that depends on the environment and the object)
To find the specific formula, we need to solve this ordinary differential equation (ODE):
dT/dt = -k(T - T_s)
b) To find the temperature of the water after 3 hours, we need to solve the ODE with the given initial condition: T(0) = 35 (initial temperature of the water)
c) To find how long it will take for the water to reach 60 degrees F, we need to solve the ODE with the initial condition T(0) = 35 and find the time when T = 60.
To calculate the values of k, T_s, and T(t) for specific times, we need additional information.