To determine which form of equation best models the price of a gallon of milk over the years, we can evaluate the fit of different models to the given data points.
The data points provided are:
- Year 0: [tex]\(\$ 3.00\)[/tex]
- Year 1: [tex]\(\$ 3.12\)[/tex]
- Year 2: [tex]\(\$ 3.24\)[/tex]
- Year 3: [tex]\(\$ 3.37\)[/tex]
- Year 4: [tex]\(\$ 3.51\)[/tex]
We consider three types of models:
1. Linear Model: [tex]\( y = ax + b \)[/tex]
2. Quadratic Model: [tex]\( y = ax^2 + b \)[/tex]
3. Exponential Model: [tex]\( y = ab^x \)[/tex]
To find the best fitting model, we compute the residuals (the sum of the squares of the differences between the observed and predicted values) for each model.
Residuals:
- The residual for the Linear Model, [tex]\( y = ax + b \)[/tex], is approximately [tex]\(0.000189999999999999\)[/tex].
- The residual for the Quadratic Model, [tex]\( y = ax^2 + b \)[/tex], is approximately [tex]\(1.1428571428569672e-05\)[/tex].
- The residual for the Exponential Model, [tex]\( y = ab^x \)[/tex], is approximately [tex]\(2.7335804358664766e-05\)[/tex].
The model with the smallest residual is the one that best fits the data:
- Linear Model: [tex]\(0.000189999999999999\)[/tex]
- Quadratic Model: [tex]\(1.1428571428569672e-05\)[/tex]
- Exponential Model: [tex]\(2.7335804358664766e-05\)[/tex]
Since the quadratic model has the smallest residual ( [tex]\(1.1428571428569672e-05\)[/tex] ), it provides the best fit.
Therefore, the equation used to model the price of a gallon of milk is:
[tex]\[ y = ax^2 + b \][/tex]
