Problem 42 - Precalculus : Building Concepts and Connections

Question

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?

Step-by-Step Solution

Answer

The specific constants \(C\) and \(a\) would be determined using the actual data points and computational measures. Then, substitute \(t = 60\) into your exponential function to find the projected temperature after 1 hour.

Step 1: Create a Scatter Plot

Plot the given table data using Time on the x-axis and Temperature on the y-axis. This gives a visual representation of the data.

Step 2: Find Exponential Function

Now, fit an exponential function \(f(t) = Ca^t\) to this data. This can be done using methods such as method of least squares. This involves finding the constants \(C\) and \(a\) that minimize the sum of the squares of the residuals (the vertical distances of the points from the curve).

Step 3: Model Verification

Verify the model obtained by superposing the curve onto the scatter plot to ensure it represents the data correctly.

Step 4: Predict Future Temperature

Using the obtained model, predict the temperature of the water after 1 hour (60 minutes), by substituting \(t = 60\) in the exponential function and solving for \(f(t)\).

Key Concepts

Scatter Plot

Creating a scatter plot is an excellent way to visualize data, especially when dealing with time series like temperature changes. Essentially, a scatter plot lets us see how one variable reacts as the other one changes, making it a great tool for modeling relationships between two sets of data.
Here is how you can create a simple scatter plot:

• Determine your axes: In this example, assign 'Time in minutes' to the x-axis and 'Temperature in Celsius' to the y-axis.
• Plot each data point given in the table on the graph.
• The result is a visual representation where you can observe the trend, such as cooling over time.

This initial visualization is crucial before creating any mathematical models, as it gives you a clear, disadvantage-free perspective of the data's overall behavior. Once plotted, patterns like linear or exponential changes become more evident.

Least Squares Method

The Least Squares Method is a mathematical approach used to find the best-fitting curve through a given set of data points. This method minimizes the sum of the squares of the residuals, or the differences between observed values and the values predicted by the model.
Here's a simple breakdown of how it works:

• Imagine each point on your scatter plot. The vertical distance from each point to the curve (e.g., an exponential function) you're trying to fit is called the residual.
• The method squares these distances to avoid negative values canceling out the positive ones. This ensures emphasis on larger deviations.
• The best-fitting curve is one that minimizes the total of these squared differences.

Using this method, we are able to determine the constants in our function, such as the exponential model in the exercise, ensuring the function fits the data as smoothly as possible. This approach is highly efficient for modeling and can be applied to various types of data relationships, not just exponential.

Temperature Change Modeling

Temperature change modeling is a fascinating application of mathematics to represent how temperature changes over time, often using functions like exponentials. In this scenario, you're tasked with identifying a model that fits the cooling pattern of a cup of hot water.
Here's how the process works:

• After creating your scatter plot and deciding on an exponential model like \(f(t) = Ca^t\), you use the Least Squares Method to define the values of constants \(C\) and \(a\).
• These constants help form a model that reflects the physical cooling process, which often follows Newton's Law of Cooling, suggesting exponential change.
• Predict future temperatures by putting the desired future time (e.g., 60 minutes for this exercise) into your model, providing a clear, data-driven prediction of the temperature at that point.

The choice of an exponential function is significant because it inherently captures the diminishing rate of change commonly seen in cooling, where changes are most rapid initially and slow over time. This approach connects mathematical insight with natural phenomena, enhancing comprehension of real-world dynamics.