how long until the temperature in the bag is within four tenths of a degree of the temperature of the water?

Question

While you are letting the filter remove more of the chlorine from the water, you are begin to acclimate your fish to the temperature of the water in the fish tank. In order to not shock the fish, you place the bag that the fish is in into the tank to let the water in the bag slowly reach the temperature of the water in the tank. The water in the bag is currently 27oC, while the tank water is 24oC. After 2 minutes the water temperature in the bag has dropped by one degree. If the rate of change of temperature of the water in the bag is proportional to the difference between the temperature of the water in the bag and the water in the tank (Newton’s Law of Heating/Cooling), how long until the temperature in the bag is within four tenths of a degree of the temperature of the water?

Answer 1

The equation for the solution to dT_B/dt = -h(T_B-T_T), where h is a constant and B stands for bag and T subscript stands for tank, is

(T_B - T_T)/(T_B(0) - T_T) = e^-ht

You can see that the bag has a T = T_B(0) at t = 0 and that it is T_T as t --> ∞

We can solve for h using the data given:

h = -1/(2 min) * ln((26-24)/(27-24))

Once you have h, solve for t by inputting T_B = 24.4. Good luck!