# a mathematician does some market research to find the effect that hourly wage has on the amount of money a repair person earns. she knows that if the repair person charges too little , lots of...

a mathematician does some market research to find the effect that hourly wage has on the amount of money a repair person earns.

she knows that if the repair person charges too little , lots of people would hire that person, but the person wouldnt earn very much because the hourly rate would be too low.

she also knows that if the repair person charges a very high hourly rate, very few customers will hire the person and, again, the repair person won't earn very much .

she believes that there must be an hourly rate that the repair person could charge that would result in the maximum rearnings.

the table below shows the average weekly earnings of repiar people based on the hourly rate they charge.

hourly rate \$       weekly earnings \$

10                           700

25                           1450

45                            1890

65                           1690

90                           540

questions

1) do you agree with the mathematician that there is probably an hourly an hourly rate will maximize the repair person's earnings , explain

2) determine te equation for the parabola of best fit for the data ,. Round the numbers in your equation to one decimal place.

3) determine the coordinates for the vertex of your parabola of best fit.

4) what is the meaning of the vertex point

5) what hourly rate will allow the repair person to make the most most money,

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

(1) Yes -- the values increase and then decrease which indicates a possible maximum.

(2) Using technology we find `y=-.8x^2+78x` to fit all of the data points.

If you cannot use technology (graphing calculator, spreadsheet, etc...) you can choose any three non-collinear points and solve a system of three equations in three unknowns. For example, using (25,1450),(45,1890), and (90,540) we get:

625a+25b+c=1450
2025a+45b+c=1890
8100a+90b+c=540

Then we can solve using substitution, linear combinations, or matrices.

(3) If the quadratic is in the form `y=ax^2+bx+c` then the x-coordinate of the vertex is `x=-b/(2a)` so we have `x=78/(-1.6)=48.75`

Substituting for x we solve for y to get y=1901.25 so the vertex is at (48.75,1901.25)

If you know calculus then `y'=-1.6x+78` and the maximum occurs when -1.6x+78=0 ==>x=48.75 as above.

(4) The vertex is the maximum point on the graph indicating the maximum earnings available.

(5) The repair person should charge \$48.75 per hour to earn a maximum of \$1901.25 per week.