Problem 50 - Algebra and Trigonometry

Question

Exercises \(47-52\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(72,\) you will use your graphing utility to obtain these functions.) Temperature Increase in an Enclosed Vehicle $$ \begin{array}{cc} {\text { Minutes }} & {\text { Temperature Increase }\left(^{\circ} \mathrm{F}\right)} \\ {10} & {19^{\circ}} \\ {20} & {29^{\circ}} \\ {30} & {34^{\circ}} \\ {40} & {38^{\circ}} \\ {50} & {41^{\circ}} \\ {60} & {43^{\circ}} \end{array} $$

Step-by-Step Solution

Answer

Based on the scatter plot, an exponential model is the best choice for modeling the temperature increase given the time passed in an enclosed vehicle.

Step 1: Represent the data as points

Convert the given table into coordinate points. The 'minutes' are the x-values and the 'Temperature Increase' are the y-values. This produces the following set of points: (10,19), (20,29), (30,34), (40,38), (50,41), (60,43).

Step 2: Create a Scatter Plot

Plot these points on an x-y graph. The x-axis will represent time in minutes and the y-axis will represent the temperature increase. This gives us a scatter plot for the data.

Step 3: Evaluate the Scatter Plot

Observe the shape and distribution of the plotted points. If they follow a straight line, it suggests a linear function. If they appear to form a curve that rises or falls rapidly at first and then levels off, it suggests an exponential function. If they form a curve that rises or falls slowly at first and then more rapidly, it suggests a logarithmic function. Here, the points seem to rise rapidly at first and then level off, so an exponential function would likely be the best fit for the data.

Key Concepts

Exponential Functions

Exponential functions are fascinating and commonly used to model growth or decay processes in various fields. They are written in the form: \[ f(x) = a \cdot b^x \]where:

• \( a \) is the initial value
• \( b \) is the base or growth factor
• \( x \) represents the independent variable

Exponential functions are ideal to use when the rate of change of a quantity is proportional to the quantity itself. A good example is population growth or radioactive decay.

For our exercise, when the temperature increase levels off gradually after an initial rapid rise, this suggests an exponential relationship. In a scatter plot, an exponential function often creates a distinct curve that quickly steepens or flattens depending on whether it represents growth or decay, respectively. By observing how the temperature increase does not directly proceed in equal steps for each minute, but rather slows down, it's indicative of an exponential pattern.

Linear Functions

Linear functions are the simplest form of functions and are characterized by a constant rate of change. The general form of a linear function is: \[ f(x) = mx + c \]where:

• \( m \) is the slope of the line
• \( c \) is the y-intercept

In simpler terms, linear functions create straight lines when plotted on a graph. They are useful for predicting relationships where the change remains consistent across intervals.

From our original exercise data, if the plotted points were to align neatly in a straight line, this would suggest a linear relationship. The temperature increase would then proceed evenly with each minute, unlike the provided dataset where increases begin quickly but then taper off. So, while linear functions are prevalent in modeling, this dataset doesn't follow a linear trend.

Logarithmic Functions

Logarithmic functions are somewhat the inverse of exponential functions and are written as:\[ f(x) = a + b \log(x) \]where:

• \( a \) shifts the graph vertically
• \( b \) stretches or compresses the graph horizontally

Logarithmic functions are perfect for situations where the rate of change decreases over time. They appear as curves that start off steep and gradually level out as \( x \) increases.

Although our exercise data demonstrates a rapid increase in temperature that slows over time, this pattern aligns more closely with an exponential rather than logarithmic trend since that increase tapered off too soon. In contrast, logarithmic growth would maintain a decreasing rate but not as sharply initially as seen here. Thus, while the data has some similarity with logarithmic functions, its characteristics still point towards an exponential curve.