Question

An object at a temperature of \(160^{\circ} \mathrm{C}\) is removed from a furnace and placed in a room at \(20^{\circ} \mathrm{C}\). The table shows the temperatures \(d\) (in degrees Celsius) at selected times \(t\) (in hours) after the object was removed from the furnace. Use a graphing calculator to find a logarithmic model of the form \(t=a+b \ln d\) that represents the data. Estimate how long it takes for the object to cool to \(50^{\circ} \mathrm{C}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{d} & 160 & 90 & 56 & 38 & 29 & 24 \\ \hline \boldsymbol{t} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{array} $$

Short answer

The solution can only be computed specifically with a calculator handy. After finding the coefficients \(a\) and \(b\) using logarithmic regression with the provided data, we plug these into our model \(t = a + b \ln d\). To find the time it takes for the object to cool to \(50^{\circ}C\), we substitute \(d=50\) into our model and solve for \(t\). The exact numerical solution will depend on the specific values of \(a\) and \(b\) obtained, and therefore is not present in this response.

Step-by-step solution

Identifying the Data Points

Firstly, identify the given data points as \(d,t\) pairs. As given in the table, these pairs are (160,0), (90,1), (56,2), (38,3), (29,4), and (24,5).

Using Logarithmic Regression

Use a graphing calculator to conduct a regression analysis. Use the logarithmic regression feature on your calculator to find the equation of the line \(t = a + b \ln d\). Enter the temperature values as your \(y\) variables and the time values as your \(x\) variables.

Determine the Constants \(a\) and \(b\)

After conducting the logarithmic regression analysis, your calculator will provide you with the constants \(a\) and \(b\). These constants are the coefficients of the logarithmic model equation \(t = a + b \ln d\).

Formulate the Logarithmic Model

With the calculated constants \(a\) and \(b\), the logarithmic model equation can be formulated by substituting these values into the equation \(t = a + b \ln d\).

Estimate the Time

Finally, to estimate when the object will cool down to \(50^{\circ} \mathrm{C}\), substitute \(d=50\) into the logarithmic model equation found earlier, and solve for \(t\). This \(t\) value will indicate the estimated time (in hours) it takes for the object to cool down to \(50^{\circ} \mathrm{C}\).

Key concepts

Logarithmic Model

Understanding the logarithmic model is essential when dealing with data that changes at a rate that slows over time. The basic form of a logarithmic model is an equation represented as t = a + b \( \ln d \), where t typically represents time, d denotes some function of time, and a and b are constants determined by the data. Essentially, this model uses the natural logarithm (ln) to transform non-linear data into a linear form which allows for easier analysis and prediction of future values.

When we apply the logarithmic model to real-world problems, such as cooling temperature, it reflects the tendency of change to be rapid at first and then slow down as the difference between the object's temperature and its surroundings decreases. This behavior suits the properties of logarithmic functions where initial large inputs lead to large outputs, but the output values grow more slowly as the input grows, much like the cooling of an object.

Graphing Calculator

To analyze and model data, graphing calculators are invaluable tools. To perform a logarithmic regression with a graphing calculator, you first input the paired data points. After entering the datasets as coordinates—where the independent variable t is plotted against the dependent variable d—you can utilize the calculator's built-in regression feature to find the best-fit logarithmic model that represents the data.

Graphing calculators will automatically calculate the constants a and b for the logarithmic equation and may also provide statistical information like the coefficient of determination (R^2) to assess the model's accuracy. Once the values are computed, they can be used to make predictions, such as estimating the time for an object to cool to a specific temperature.

Cooling Temperature Problem

A cooling temperature problem involves predicting how long it will take for an object to reach a certain temperature after it has been removed from a source of heat, as it equilibrates with its environment. This kind of problem is common in fields such as physics, engineering, and environmental science. The goal is to determine the relationship between time and temperature as the object cools.

The cooling process often follows patterns that are well-characterized by logarithmic relationships. This is due to the law of cooling, which states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature. Using a logarithmic model to represent this relationship allows us to make predictions about an object's temperature at any given time point.

Data Analysis

Data analysis is an imperative step when working with any form of empirical information. It involves processing and reviewing raw data to extract meaningful insights. In the context of logarithmic regression, data analysis transforms the chaotic nature of real-world data into a form where patterns and relationships can be identified and used to make predictions.

Starting with a collection of data points, analysts use logarithmic regression to discern the underlying relationship between variables. Good data analysis also includes evaluating the strength of the model through statistical measures, determining its reliability, and using it to make accurate forecasts. Clear and concise interpretations are crucial for making informed decisions in various scientific and practical applications.