Calculus Word Problem?
In this question we use Newton’s Law of heating to find a temperature function.
Suppose that a container of milk is removed from the refrigerator and
placed on a table. The temperature of the room is constantly E and we let
T(t) denote the temperature (in degrees Celsius) of the milk after t minutes.
1.Write an equation which corresponds to the description: “The rate of
change of the temperature of the milk is proportional to the difference
between the room temperature and the milk temperature”. Introduce
new constant(s) as needed.
2.Define a new function of time S by the rule S(t) = E − T(t). How do dS/dt and dT/dt relate?
3. Use 1. and 2. to determine a differential equation (not using T) which
is satisfied by S.
4. What is the general form of S?
5. What is the general form of T?
6. Assume that E = 20◦ C and suppose that the initial temperature of the
milk is 5◦ C and the temperature of the milk after 10 minutes is 10◦ C
degrees. What is the function T(t)? (keep constants to 4 decimal places)
In this question we use Newton’s Law of heating to find a temperature function.
Suppose that a container of milk is removed from the refrigerator and
placed on a table. The temperature of the room is constantly E and we let
T(t) denote the temperature (in degrees Celsius) of the milk after t minutes.
1.Write an equation which corresponds to the description: “The rate of
change of the temperature of the milk is proportional to the difference
between the room temperature and the milk temperature”. Introduce
new constant(s) as needed.
2.Define a new function of time S by the rule S(t) = E − T(t). How do dS/dt and dT/dt relate?
3. Use 1. and 2. to determine a differential equation (not using T) which
is satisfied by S.
4. What is the general form of S?
5. What is the general form of T?
6. Assume that E = 20◦ C and suppose that the initial temperature of the
milk is 5◦ C and the temperature of the milk after 10 minutes is 10◦ C
degrees. What is the function T(t)? (keep constants to 4 decimal places)
1 Answer
Explanation:
Room Temp =
Milk Temp =
1. The Equation
2. The Relationship
3. The DE
Subs result from (2) into result from from (1) we get:
4. Solve for S
The DE from (3) is a First Order Separable DE, and we can collect terms as follows;
Separating the variables we get:
And Integrating gives us:
5. Solve for T
6. Apply Initial Conditions
So we have:
And so: