An object is heated to 90 degrees Celcius and is then allowed to cool in a room whose air temperature is 25 degrees celcius. If the temperature of the object is 75 degrees celcius after 10 minutes, when will its temperature be 50 degrees celcius?

Question

I have no idea how to do this. There are just to many numbers involved!!

Answer 1

Don't worry! Let me guide you through this problem. We can use Newton's Law of Cooling, which states:

dT/dt = -k(T - T_env)

Where:
- dT/dt is the rate of change in the object's temperature with respect to time,
- k is a cooling constant (depends on the object and medium, but remains constant for this problem),
- T is the object's temperature at a given time,
- T_env is the temperature of the environment (in this case, room air temperature).

First, we'll find the cooling constant k. To do that, we can use the information that the object's temperature is 75° C after 10 minutes (10 minutes = 600 seconds).

We know that:
T = 75° C
T_env = 25° C
dT/dt = (75 - 90) / 600 = -15 / 600 = -1/40 °C/s

Now, we can set up the equation:

-1/40 = -k(75 - 25)

Solve for k:

k = (1/40) / 50 = 1/2000 (unit doesn't matter here as it will cancel out)

Now that we have the cooling constant (k), we can find the time it takes for the object to reach 50° C.

Let's call that time t_seconds. Now we need to set up the equation with the information we have:

dT/dt = -k(T - T_env)

We have:
T = 50° C
T_env = 25° C
k = 1/2000

Now we'll find dT/dt when the temperature of the object is 50° C:

dT/dt = (50 - 90) / t_seconds

Inserting the values into the equation:

(50 - 90) / t_seconds = -(1/2000) * (50 - 25)

Solving for t_seconds:

-40 / t_seconds = -1/80

t_seconds = 40 * 80 = 3200 seconds

Now convert the seconds to minutes:

3200 seconds = 3200 / 60 = 53.33 minutes

So, it will take approximately 53.33 minutes for the object's temperature to reach 50° C.