- Define the quadratic model \(f(x) = ax^2 + bx + c\) to fit the given data.
- Use the method of least squares to find the coefficients \(a \approx 0.8489\), \(b \approx -17.7418\), and \(c \approx 190.1179\).
- Calculate the predicted cost for \(x=2\) by evaluating \(f(2) = 0.8489(2)^2 - 17.7418(2) + 190.1179 \approx 158.03\).
- Round the final result to the nearest dollar, yielding \(\boxed{158}\) dollars.
Explanation
- Problem Analysis
To determine the quadratic curve of best fit, we model the cost as a function \(f(x) = ax^2 + bx + c\), where \(x\) is the number of TVs produced and \(f(x)\) is the per-item cost in dollars. We are given the following data points: \((3, 151.1)\), \((5, 118.47)\), \((6, 103.07)\), \((9, 99.09)\), \((10, 116.24)\), and \((12, 89.67)\). Our objective is to find the coefficients \(a, b,\) and \(c\) that minimize the sum of squared errors and then predict the cost for \(x=2\).
- Determining the Quadratic Model
Using the method of least squares for quadratic regression, we calculate the coefficients \(a, b,\) and \(c\) for the function \(f(x) = ax^2 + bx + c\). Applying this method to the given data points, we find:
\[a \approx 0.8489\]
\[b \approx -17.7418\]
\[c \approx 190.1179\]
Thus, the quadratic model of best fit is:
\[f(x) = 0.8489x^2 - 17.7418x + 190.1179\]
- Predicting the Cost and Final Conclusion
Now that we have the model, we calculate the predicted per-item cost for \(x = 2\) TVs:
\[f(2) = 0.8489(2)^2 - 17.7418(2) + 190.1179\]
\[f(2) = 0.8489(4) - 35.4836 + 190.1179\]
\[f(2) = 3.3956 - 35.4836 + 190.1179\]
\[f(2) = 158.0299\]
Rounding this result to the nearest dollar, we get \(\$158\).
Examples
In business and economics, quadratic regression is frequently used to model costs that do not decrease linearly. For instance, a manufacturer might experience high initial costs due to setup and low volume, then achieve economies of scale as production increases, but eventually face rising marginal costs due to overtime pay or equipment maintenance as production reaches maximum capacity. By modeling these trends with a quadratic function, business owners can predict costs at various production levels and plan their inventory and budget more effectively.