Question

(Quadratic Regressions) 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 profitearned at that price, y. Write a quadratic regression equation for this set of data,rounding all coefficients to the nearest tenth. Using this equation, find the profit, tothe nearest dollar, for a selling price of 27.5 dollars.

Answer 1

`\begin{gathered} A\text{ quadratic equation is given by;} \\ y=ax^2+bx+c \\ \text{From the table, } \\ 386=(13.25)^2a+13.25b+c \\ 386=175.5625a+13.25b+c\ldots\ldots\ldots\ldots\ldots\ldots...........\ldots\text{equation 1} \end{gathered}`
`\begin{gathered} \text{similarly;} \\ 476=(15.50)^2a+15.50b+c \\ 476=240.25a+15.50b+c\ldots\ldots\ldots\ldots\ldots\ldots\text{...........equation 2} \end{gathered}`
`\begin{gathered} 611=(21.50)^2a+21.50b+c \\ 611=462.25a+21.50b+c\ldots\ldots\ldots\ldots\ldots\text{.........equation 3} \end{gathered}`

Solving the simultaneous of three unknowns;

`\begin{gathered} a=-1.7,\text{ b=87.5 and c=-478}.3 \\ \text{Thus;} \\ y=-1.7x^2+87.5x-478.3 \end{gathered}`
`\begin{gathered} \text{When x=27.5dollars;} \\ y=-1.7(27.5)^2+87.5(27.5)-478.3 \\ y=642.3 \end{gathered}`

The profit for the price of 27.5 dollars is 642.3