To determine whether the data for the estimated number of deer living in a forest over a five-year period is best represented by a linear, exponential, or quadratic model, we will fit each type of model to the data and compare the residuals (the differences between the observed and the predicted values). The model with the smallest residuals will be the best fit.
Let's start by summarizing the given data:
- Year 0: Population 98
- Year 1: Population 77
- Year 2: Population 61
- Year 3: Population 48
- Year 4: Population 38
### Linear Model
To fit a linear model, we assume a relationship of the form:
[tex]\[ P(t) = a + bt \][/tex]
where [tex]\( t \)[/tex] is the year and [tex]\( P(t) \)[/tex] is the population.
### Quadratic Model
To fit a quadratic model, we assume a relationship of the form:
[tex]\[ P(t) = a + bt + ct^2 \][/tex]
where [tex]\( t \)[/tex] is the year and [tex]\( P(t) \)[/tex] is the population.
### Exponential Model
To fit an exponential model, we assume a relationship of the form:
[tex]\[ P(t) = a \cdot e^{bt} \][/tex]
where [tex]\( t \)[/tex] is the year and [tex]\( P(t) \)[/tex] is the population.
### Comparing Residuals
After fitting each of these models to the data, we calculate the sum of the squared residuals for each model. The residual is the difference between the observed population and the predicted population from the model.
The sum of squared residuals for each model is determined as follows:
1. Linear Model: The sum of squared residuals is approximately [tex]\( 45.10 \)[/tex].
2. Quadratic Model: The sum of squared residuals is approximately [tex]\( 0.46 \)[/tex].
3. Exponential Model: The sum of squared residuals is approximately [tex]\( 0.09 \)[/tex].
### Best Fit Model
The model with the smallest residuals provides the best fit to the data. In this case, the exponential model has the smallest sum of squared residuals.
### Exponential Model Equation
Given that the exponential model is the best fit, the form of the equation is:
[tex]\[ P(t) = a \cdot e^{bt} \][/tex]
The parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex] have been determined to provide the best fit:
- [tex]\( a \approx 98.0 \)[/tex]
- [tex]\( b \approx -0.20 \)[/tex]
Thus, the exponential model equation that best represents the data is:
[tex]\[ P(t) = 98.0 \cdot e^{-0.20t} \][/tex]
This equation models the declining deer population over the five-year period.
