Questions: Company X tried selling widgets at various prices to see how much profit they would make. The following 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. Using this equation, find the profit, to the nearest dollar, for a selling price of 36.25 dollars. Price (x) Profit (y) 12.25 16216 14.75 24134 20.00 34589 28.50 37621 33.75 30737 36.25 24424 Copy Values for Calculator Open Statistics Calculator Answer Attempt 1 of 3

Solution Steps

The data provided consists of selling prices \( x \) and corresponding profits \( y \):

\[ \begin{array}{|c|c|} \hline x & y \\ \hline 12.25 & 16216 \\ 14.75 & 24134 \\ 20.00 & 34589 \\ 28.50 & 37621 \\ 33.75 & 30737 \\ 36.25 & 24424 \\ \hline \end{array} \]

Using quadratic regression, we derive the equation of the form:

\[ y = ax^2 + bx + c \]

The coefficients obtained are:

\[ a = -125.05, \quad b = 6409.89, \quad c = -43432.81 \]

Thus, the quadratic regression equation is:

\[ y = -125.05x^2 + 6409.89x - 43432.81 \]

To find the predicted profit for a selling price of \( x = 36.25 \):

\[ y = -125.05(36.25)^2 + 6409.89(36.25) - 43432.81 \]

Calculating this gives:

\[ y \approx 24602 \]

Final Answer