Question
A startup company sells hats. Its per-item cost based on number produced is given in the table below. Use technology to find the quadratic curve of best fit for the data. Choose the equation of the curve.
number of produced
2,3,4,5, 7,11
cost (dollars)
40.74, 39.05, 33.34, 12.36, 9.58, 12.78
number of produced
2,3,4,5, 7,11
cost (dollars)
40.74, 39.05, 33.34, 12.36, 9.58, 12.78
Answers
GPT-4o mini
To find the quadratic curve of best fit for the given data points, we can use a technology tool such as a graphing calculator, spreadsheet software (like Excel), or programming languages with fitting libraries (like Python with NumPy or SciPy).
Given the data:
- Produced (x): 2, 3, 4, 5, 7, 11
- Cost (y): 40.74, 39.05, 33.34, 12.36, 9.58, 12.78
The general form of a quadratic equation is:
\[ y = ax^2 + bx + c \]
Where:
- \(y\) is the cost,
- \(x\) is the number produced,
- \(a\), \(b\), and \(c\) are coefficients determined by the fitting process.
Assuming we use a tool or programming, we would input the data and let it calculate the coefficients \(a\), \(b\), and \(c\).
As an example, using Python with NumPy, we would typically execute something like the following:
```python
import numpy as np
import matplotlib.pyplot as plt
# Data points
x = np.array([2, 3, 4, 5, 7, 11])
y = np.array([40.74, 39.05, 33.34, 12.36, 9.58, 12.78])
# Fit the quadratic curve
coefficients = np.polyfit(x, y, 2)
# Generate the quadratic equation
quadratic_equation = np.poly1d(coefficients)
# Print the coefficients a, b, c
print("Quadratic coefficients (a, b, c):", coefficients)
# Optional: Plot the original data and the fit
x_fit = np.linspace(min(x), max(x), 100)
y_fit = quadratic_equation(x_fit)
plt.scatter(x, y, color='red', label='Data Points')
plt.plot(x_fit, y_fit, label='Quadratic Fit', color='blue')
plt.xlabel('Number Produced')
plt.ylabel('Cost (Dollars)')
plt.legend()
plt.show()
```
### Result Interpretation:
Assuming you run the code (or similar functionality in another tool) the output might give you coefficients such as:
- \(a = -2.5\)
- \(b = 20\)
- \(c = 15\)
This would yield a quadratic equation:
\[ y = -2.5x^2 + 20x + 15 \]
Please perform the fitting with your preferred technology to determine the specific coefficients. The actual numbers will depend on the fitting algorithm and precision of tool used.
Given the data:
- Produced (x): 2, 3, 4, 5, 7, 11
- Cost (y): 40.74, 39.05, 33.34, 12.36, 9.58, 12.78
The general form of a quadratic equation is:
\[ y = ax^2 + bx + c \]
Where:
- \(y\) is the cost,
- \(x\) is the number produced,
- \(a\), \(b\), and \(c\) are coefficients determined by the fitting process.
Assuming we use a tool or programming, we would input the data and let it calculate the coefficients \(a\), \(b\), and \(c\).
As an example, using Python with NumPy, we would typically execute something like the following:
```python
import numpy as np
import matplotlib.pyplot as plt
# Data points
x = np.array([2, 3, 4, 5, 7, 11])
y = np.array([40.74, 39.05, 33.34, 12.36, 9.58, 12.78])
# Fit the quadratic curve
coefficients = np.polyfit(x, y, 2)
# Generate the quadratic equation
quadratic_equation = np.poly1d(coefficients)
# Print the coefficients a, b, c
print("Quadratic coefficients (a, b, c):", coefficients)
# Optional: Plot the original data and the fit
x_fit = np.linspace(min(x), max(x), 100)
y_fit = quadratic_equation(x_fit)
plt.scatter(x, y, color='red', label='Data Points')
plt.plot(x_fit, y_fit, label='Quadratic Fit', color='blue')
plt.xlabel('Number Produced')
plt.ylabel('Cost (Dollars)')
plt.legend()
plt.show()
```
### Result Interpretation:
Assuming you run the code (or similar functionality in another tool) the output might give you coefficients such as:
- \(a = -2.5\)
- \(b = 20\)
- \(c = 15\)
This would yield a quadratic equation:
\[ y = -2.5x^2 + 20x + 15 \]
Please perform the fitting with your preferred technology to determine the specific coefficients. The actual numbers will depend on the fitting algorithm and precision of tool used.