To determine which function best models the data provided in the table for the estimated number of bees [tex]\( y \)[/tex] in a hive after [tex]\( x \)[/tex] days, we compare different models and evaluate their respective errors. Here are the proposed functions:
1. [tex]\( y = 9958 \cdot (0.972)^x \)[/tex]
2. [tex]\( y = 0.972 \cdot (9958)^x \)[/tex]
3. [tex]\( y = 9219 \cdot x - 150 \)[/tex]
4. [tex]\( y = -150 \cdot x + 9219 \)[/tex]
To make this comparison, we calculate the squared errors for each model. The squared error measures how much the estimated values deviate from the actual values in the data set. Smaller errors indicate a better fit.
Given the results for the errors:
1. [tex]\( y = 9958 \cdot (0.972)^x \)[/tex] has a squared error of [tex]\( 5950.85 \)[/tex]
2. [tex]\( y = 0.972 \cdot (9958)^x \)[/tex] has a squared error of approximately [tex]\( 6.20218510682898e+399 \)[/tex]
3. [tex]\( y = 9219 \cdot x - 150 \)[/tex] has a squared error of [tex]\( 456917467500 \)[/tex]
4. [tex]\( y = -150 \cdot x + 9219 \)[/tex] has a squared error of [tex]\( 1774566 \)[/tex]
Analyzing the errors:
- The exponential model [tex]\( y = 0.972 \cdot (9958)^x \)[/tex] has an extremely large error, indicating it does not fit the data well at all.
- The linear models [tex]\( y = 9219 \cdot x - 150 \)[/tex] and [tex]\( y = -150 \cdot x + 9219 \)[/tex] also have significantly larger errors compared to the first model.
- The exponential model [tex]\( y = 9958 \cdot (0.972)^x \)[/tex] has the smallest error among all the models.
Therefore, the function that best models the data is:
[tex]\[ y = 9958 \cdot (0.972)^x \][/tex]
This function provides the closest estimates to the actual number of bees in the hive over time, as evidenced by the lowest squared error.
