The following table gives the temperature, in degrees...

The following table gives the temperature, in degrees Celsius, of a cup of hot water sitting in a room with constant temperature. The data was collected over a period of 30 minutes. (Source: www.phys. unt.edu, Dr. James A. Roberts)$$\begin{array}{|c|c|} \hline\text { Time } & \text { Temperature } \\(\mathrm{min}) & (\text { degrees Celsius }) \\ \hline0 & 95 \\1 & 90.4 \\5 & 84.6 \\10 & 73 \\15 & 64.7 \\20 & 59 \\25 & 54.5 \\29 & 51.4\\\hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form $f(t)=C a^{2}$ that best fits the data. Let $t$ be the number of minutes the water has been cooling. (b) Using your modicl, what is the projected temperature of the water after 1 hour?

The following table gives the temperature, in degrees Celsius, of a cup of hot water sitting in a room with constant temperature. The data was collected over a period of 30 minutes. (Source: www.phys. unt.edu, Dr. James A. Roberts)$$\begin{array}{|c|c|} \hline\text { Time } & \text { Temperature } \\(\mathrm{min}) & (\text { degrees Celsius }) \\ \hline0 & 95 \\1 & 90.4 \\5 & 84.6 \\10 & 73 \\15 & 64.7 \\20 & 59 \\25 & 54.5 \\29 & 51.4\\\hline\end{array}$$
(a) Make a scatter plot of the data and find the exponential function of the form $f(t)=C a^{2}$ that best fits the data. Let $t$ be the number of minutes the water has been cooling.
(b) Using your modicl, what is the projected temperature of the water after 1 hour?

Key Concepts

Extrapolation

Extrapolation is the process of estimating values beyond the range of observed data using a fitted model. Once an accurate exponential model is established, it can be used to predict the temperature at time points not originally measured, such as forecasting the water's temperature after one hour. This predictive feature is a key benefit of developing a reliable mathematical model.

Exponential Function Modeling

An exponential function of the form f(t) = C · a^t is used to mathematically model processes where change happens at a constant percentage rate over time. This form is particularly effective in modeling the cooling of an object, where 'C' represents the initial condition and 'a' the base corresponding to the decay factor per time unit. Determining the values of these constants based on the data is crucial for predictive analysis.

Curve Fitting

Curve fitting is the procedure of constructing a curve, or mathematical function, that has the best fit to a series of data points. For an exponential model, this often involves transforming the data (e.g., taking logarithms) and using linear regression techniques to determine the parameters of the model. This method ensures that the function accurately represents the observed behavior of the system over the measured time period.

Exponential Decay

Exponential decay describes a process where the rate of decrease of a quantity is proportional to the current amount of the quantity itself. In the context of cooling, the temperature of the hot water decreases rapidly at first and then gradually levels off as it approaches room temperature. The exponential model captures this behavior and forms the basis for the curve fitting in this problem.

Scatter Plot

A scatter plot is a graphical tool used to display the relationship between two variables—in this case, time and temperature. Plotting the data points allows one to visually assess the pattern or trend in the data, which is essential before applying any mathematical model. It helps in identifying the type of function that may best describe the underlying relationship between the variables.

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