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.
Dirt Cheap The Cherry Hill Construction company in Branford, CT sells screened topsoil by the “yard,” which is actually a cubic yard. Let the variable x be the length (yd) of each side of a cube of screened topsoil. The table below lists the values of x along with the corresponding cost (dollars).

Verified step by step guidance

1. Step 1: Begin by plotting the given data points on a scatterplot. Use the variable x (length of each side of the cube in yards) as the independent variable on the x-axis and the cost (in dollars) as the dependent variable on the y-axis.
2. Step 2: Observe the pattern of the data points on the scatterplot. Determine whether the relationship between x and cost appears to be linear, quadratic, logarithmic, exponential, or power-based. Look for trends such as curvature, rapid growth, or proportional scaling.
3. Step 3: Fit each of the potential models (linear, quadratic, logarithmic, exponential, and power) to the data using regression techniques. For example, use the least squares method to calculate the coefficients for each model.
4. Step 4: Evaluate the goodness-of-fit for each model using statistical measures such as the coefficient of determination (R²). The model with the highest R² value is likely the best fit for the data.
5. Step 5: Once the best-fitting model is identified, write down its mathematical equation. Ensure that the model is only used within the scope of the given data, as extrapolation beyond the provided range may lead to inaccuracies.

Key Concepts

Scatterplot

A scatterplot is a graphical representation of two variables, where each point represents an observation in the dataset. In this context, the x-axis represents the length of the cube of topsoil, while the y-axis represents the corresponding cost. Analyzing the scatterplot helps identify the relationship between the variables and the potential mathematical model that fits the data.

Mathematical Models

Mathematical models are equations that describe the relationship between variables. In this exercise, you are tasked with identifying the best-fitting model from linear, quadratic, logarithmic, exponential, and power models. Each model has distinct characteristics and is suitable for different types of data patterns, making it essential to choose the one that accurately represents the observed relationship.

Cost and Volume Relationship

The relationship between cost and volume in this scenario is influenced by the cubic nature of the topsoil. As the length of each side of the cube increases, the volume—and consequently the cost—grows at a rate that may not be linear. Understanding this relationship is crucial for selecting the appropriate mathematical model that reflects how cost escalates with increasing volume.