Find a mathematical model for the verbal statement. (Use k for the constant of proportionality.)
The rate of change R of the temperature of an object is directly proportional to the difference between the temperature T of the object and the temperature Te of the environment.
wait - since this does not appear to be for a calculus class, Let's just say
R = k(T-Te)
dT/dt = k(T-Te)
Sure thing! Let's put on our mathematical hats and get to work.
According to the verbal statement, we can write the mathematical model as:
R = k(T - Te)
Here, R represents the rate of change of the temperature, T is the temperature of the object, Te is the temperature of the environment, and k is the constant of proportionality.
Now, let's add a little humor to the mix. Why did the mathematical model go to the comedy club? Because it wanted to find its constant "k"!
To find a mathematical model for the given verbal statement, we can use the following template for direct variation:
R = k(T - Te)
Here, R represents the rate of change of temperature, T represents the temperature of the object, Te represents the temperature of the environment, and k is the constant of proportionality.
To find a mathematical model for the given verbal statement, let's break it down.
The rate of change R of the temperature of an object is directly proportional to the difference between the temperature T of the object and the temperature Te of the environment.
In this scenario, "directly proportional" means that if the difference between the object's temperature and the environment's temperature increases, the rate of change of the object's temperature will also increase. It also implies that if we double the difference, the rate of change will also double.
Let's represent the constant of proportionality as k.
Based on the verbal statement, we can write the following equation:
R = k(T - Te)
Here, R represents the rate of change of temperature, T represents the temperature of the object, and Te represents the temperature of the environment.
By multiplying k with the difference (T - Te), we account for the direct proportionality between the rate of change and the temperature difference.
Therefore, the mathematical model for the given verbal statement is:
R = k(T - Te)