Answer :
- Fit a quadratic model to the data using least squares: $y = -0.00805x^2 + 0.20729x + 51.0559$.
- Fit a cubic model to the data using least squares: $y = -0.000825x^3 + 0.03065x^2 - 0.24568x + 52.0754$.
- Calculate the R-squared value for both models. The cubic model has an R-squared value of 1.0, indicating a perfect fit.
- The function that models the diver's earnings is: $\boxed{y = -0.000825x^3 + 0.03065x^2 - 0.24568x + 52.0754}$
### Explanation
1. Analyzing the Data
We are given a table of data relating the number of days a diver dives and the diver's earnings in dollars. The goal is to find a function that models the diver's earnings based on the number of days they dive. The data points are (2, 51.7), (5, 51.51), (15, 52.50), and (30, 50.00). Since the data doesn't appear linear, we'll explore quadratic and cubic models to fit the data.
2. Fitting a Quadratic Model
Let's fit a quadratic model of the form $y = ax^2 + bx + c$ to the data, where $x$ is the number of days diving and $y$ is the earnings. Using the method of least squares, we find the coefficients to be approximately $a = -0.00805$, $b = 0.20729$, and $c = 51.0559$. This gives us the quadratic model: $y = -0.00805x^2 + 0.20729x + 51.0559$.
3. Fitting a Cubic Model
Next, let's fit a cubic model of the form $y = ax^3 + bx^2 + cx + d$ to the data. Again, using the method of least squares, we find the coefficients to be approximately $a = -0.000825$, $b = 0.03065$, $c = -0.24568$, and $d = 52.0754$. This gives us the cubic model: $y = -0.000825x^3 + 0.03065x^2 - 0.24568x + 52.0754$.
4. Comparing the Models
To determine which model fits the data better, we calculate the R-squared value for both models. The R-squared value for the quadratic model is approximately 0.9278, while the R-squared value for the cubic model is 1.0. Since the cubic model has an R-squared value of 1.0, it perfectly fits the given data points.
5. Final Answer
Therefore, the cubic model provides a better fit for the given data. The function that models the diver's earnings as a function of the number of days diving is: $\boxed{y = -0.000825x^3 + 0.03065x^2 - 0.24568x + 52.0754}$
### Examples
Understanding how earnings change with the number of days worked is crucial in many professions. For example, a freelancer might use this type of analysis to model their income based on the number of days they work per month. By fitting a curve to their historical data, they can predict future earnings and make informed decisions about their workload and pricing strategies. This approach can also be used in sales to model revenue based on the number of sales calls made or in manufacturing to model production output based on the number of hours worked.
- Fit a cubic model to the data using least squares: $y = -0.000825x^3 + 0.03065x^2 - 0.24568x + 52.0754$.
- Calculate the R-squared value for both models. The cubic model has an R-squared value of 1.0, indicating a perfect fit.
- The function that models the diver's earnings is: $\boxed{y = -0.000825x^3 + 0.03065x^2 - 0.24568x + 52.0754}$
### Explanation
1. Analyzing the Data
We are given a table of data relating the number of days a diver dives and the diver's earnings in dollars. The goal is to find a function that models the diver's earnings based on the number of days they dive. The data points are (2, 51.7), (5, 51.51), (15, 52.50), and (30, 50.00). Since the data doesn't appear linear, we'll explore quadratic and cubic models to fit the data.
2. Fitting a Quadratic Model
Let's fit a quadratic model of the form $y = ax^2 + bx + c$ to the data, where $x$ is the number of days diving and $y$ is the earnings. Using the method of least squares, we find the coefficients to be approximately $a = -0.00805$, $b = 0.20729$, and $c = 51.0559$. This gives us the quadratic model: $y = -0.00805x^2 + 0.20729x + 51.0559$.
3. Fitting a Cubic Model
Next, let's fit a cubic model of the form $y = ax^3 + bx^2 + cx + d$ to the data. Again, using the method of least squares, we find the coefficients to be approximately $a = -0.000825$, $b = 0.03065$, $c = -0.24568$, and $d = 52.0754$. This gives us the cubic model: $y = -0.000825x^3 + 0.03065x^2 - 0.24568x + 52.0754$.
4. Comparing the Models
To determine which model fits the data better, we calculate the R-squared value for both models. The R-squared value for the quadratic model is approximately 0.9278, while the R-squared value for the cubic model is 1.0. Since the cubic model has an R-squared value of 1.0, it perfectly fits the given data points.
5. Final Answer
Therefore, the cubic model provides a better fit for the given data. The function that models the diver's earnings as a function of the number of days diving is: $\boxed{y = -0.000825x^3 + 0.03065x^2 - 0.24568x + 52.0754}$
### Examples
Understanding how earnings change with the number of days worked is crucial in many professions. For example, a freelancer might use this type of analysis to model their income based on the number of days they work per month. By fitting a curve to their historical data, they can predict future earnings and make informed decisions about their workload and pricing strategies. This approach can also be used in sales to model revenue based on the number of sales calls made or in manufacturing to model production output based on the number of hours worked.
