Finding the Best Model
In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.
Landing on the Moon When the Apollo spacecraft landed on the Moon, the rocket engine would typically cut off at about 1.3 meters above the surface so that hot gases and dust and other surface materials would not cause damage. The landing module was in freefall starting at about 1 meter above the surface. The table below lists the time t (seconds) after being dropped and the distance d (meters) travelled by an object dropped near the surface of the Moon.

Verified step by step guidance

1. Step 1: Construct a scatterplot by plotting the given data points with time \(t\) on the x-axis and distance \(d\) on the y-axis. This visual representation will help identify the pattern or trend in the data.
2. Step 2: Observe the shape of the scatterplot to hypothesize which type of model might fit best. For example, if the points form a straight line, a linear model might be appropriate; if the points curve upward or downward, consider quadratic, exponential, logarithmic, or power models.
3. Step 3: Test each candidate model by fitting the data to the corresponding mathematical equation: - Linear: \(d = a t + b\) - Quadratic: \(d = a t^2 + b t + c\) - Exponential: \(d = a e^{b t}\) - Logarithmic: \(d = a \ln(t) + b\) - Power: \(d = a t^b\)
4. Step 4: Calculate or estimate the parameters (\(a\), \(b\), \(c\)) for each model using methods such as least squares regression or transformation of variables (e.g., taking logarithms for exponential or power models).
5. Step 5: Compare the goodness of fit for each model by examining residuals, \(R^2\) values, or other fit statistics to determine which model best describes the data within the given range of \(t\).

Key Concepts

Scatterplot Construction

A scatterplot is a graphical representation of data points on a coordinate plane, showing the relationship between two variables. Plotting time (t) on the x-axis and distance (d) on the y-axis helps visualize the pattern or trend in the data, which is essential for selecting an appropriate model.

Types of Mathematical Models

Common models include linear, quadratic, logarithmic, exponential, and power models. Each describes different relationships: linear shows constant rate, quadratic shows acceleration, logarithmic grows slowly, exponential grows rapidly, and power models follow a specific polynomial form. Choosing the best fit depends on the data pattern.

Model Fitting and Interpretation

Model fitting involves matching a mathematical equation to the data points to describe their relationship accurately. The best model minimizes errors and reflects the physical context, such as freefall motion on the Moon, where distance often relates to time squared, suggesting a quadratic model.