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To solve this, we need to determine the quadratic regression equation from the given data and use it to predict the profit for a selling price of \( x = 45.5 \).

Here are the steps to solve:

1. **Set up the quadratic regression equation**:
   A quadratic regression equation is in the form:
   \[
   y = ax^2 + bx + c
   \]
   where \( y \) is the profit and \( x \) is the selling price. The coefficients \( a \), \( b \), and \( c \) will be determined by fitting the data.

2. **Input data points**:
   The table gives the following points:
   \[
   (11.00, 972), (13.75, 1470), (19.50, 2228), (27.00, 2754), (33.50, 2391)
   \]

3. **Perform quadratic regression**:
   Using a regression calculator or software, compute the coefficients \( a \), \( b \), and \( c \). These values will be rounded to the nearest hundredth.

4. **Substitute \( x = 45.5 \)**:
   Once the regression equation is determined, substitute \( x = 45.5 \) to calculate the predicted profit.

Let me calculate the quadratic regression coefficients and the predicted profit. numpy as np
import pandas as pd
from numpy.polynomial.polynomial import Polynomial

# Data points
x = np.array([11.00, 13.75, 19.50, 27.00, 33.50])
y = np.array([972, 1470, 2228, 2754, 2391])

# Perform quadratic regression
coefficients = np.polyfit(x, y, 2)  # np.polyfit gives coefficients in descending order

# Format coefficients to nearest hundredth
a, b, c = np.round(coefficients, 2)

# Regression equation
regression_equation = f"y = {a}x^2 + {b}x + {c}"

# Predict profit for x = 45.5
x_pred = 45.5
y_pred = np.polyval(coefficients, x_pred)
y_pred_rounded = round(y_pred)  # Round to the nearest dollar

regression_equation, y_pred_roundedThe quadratic regression equation is:

\[
y = -6.59x^2 + 358.96x - 2205.67
\]

Using this equation, the predicted profit for a selling price of \( x = 45.5 \) dollars is approximately **477 dollars**.

Would you like a breakdown of the calculations or have further questions? Here are five related questions to expand on this:

1. How do you perform quadratic regression manually?
2. What tools or calculators are best for quadratic regression?
3. Can quadratic regression be used for larger datasets, and how does it compare to linear regression?
4. How sensitive are predictions to rounding coefficients?
5. What are other use cases for quadratic regression in business?

**Tip:** Always double-check regression results by plotting the data and regression curve to verify the fit visually.

Company X tried selling widgets at various prices to see how much profit they would make. The table shows the widget selling price (x) and the total profit earned at that price (y). Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest hundredth. Find the profit, to the nearest dollar, for a selling price of 45.5 dollars.

Learn how to write a quadratic regression equation for widget sales data and predict profit at a selling price of 45.5 dollars. This step-by-step guide explains regression analysis, coefficient rounding, and profit prediction. Suitable for Grades 10-12.

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