Answer:
[tex]y=\boxed{-14.380}\;x^2+\boxed{299.970}\;x+\boxed{-950.851}[/tex]
$39
Step-by-step explanation:
Create 5 additional columns on the given table for x², x³, x⁴, xy, and x²y.
Calculate the values and add the sum of each column.
(See attachment 1).
Use the matrix equation for determining the parabolic curve, entering the sums from attachment 1:
[tex]\displaystyle a \sum x{_i}^4+b \sum x{_i}^3+c \sum x{_i}^2=\sum x{_i}^2y_i[/tex]
[tex]\implies 74446.015625a+6046.90625b+520.25c=248662.875[/tex]
[tex]\displaystyle a \sum x{_i}^3+b \sum x{_i}^2+c \sum x{_i}=\sum x{_i}y_i[/tex]
[tex]\implies 6046.90625a+520.25b+48.5c=22987.5[/tex]
[tex]\displaystyle a \sum x{_i}^2+b \sum x{_i}+c n_i=\sum y_i[/tex]
[tex]\implies 520.25a+48.5b+5c=2313[/tex]
Solve for a, b and c using a calculator:
- a = -14.38019324...
- b = 299.9701548...
- c = -950.8513949...
Insert these values (rounding to 3 decimal places) into the quadratic formula [tex]y=ax^2+bx+c[/tex] :
[tex]\large\boxed{y=-14.380x^2+299.970x-950.851}[/tex]
To find the profit (y) for a selling price of 16.75, substitute x = 16.75 into the found quadratic equation:
[tex]\implies y=-14.380(16.75)^2+299.970(16.75)-950.851[/tex]
[tex]\implies y=39.157...[/tex]
Therefore, the profit is $39 to the nearest dollar.
Note: Attachment 2 is the two-variable regression analysis as calculated by an online graphing calculator, proving that the y-value is 39 (nearest whole number) when x=16.75.
