Quadratic model for the total cost - using derivatives

[kjones]

Please help me figure out this:

The data in the table gives the total daily cost "y" (in $) of producing "x" dozen donuts at different production levels.

|----------------|-----|-----|-----|-----|-----|-----|
| donuts (doz.)  |  50 | 100 | 150 | 200 | 250 | 300 |
| ---------------|-----|-----|-----|-----|-----|-----|
| total cost ($) | 119 | 187 | 248 | 382 | 505 | 695 |
|----------------|-----|-----|-----|-----|-----|-----|

a) Find a quadratic model for the total cost.

b) Find an equation for the average cost.

c) Find the production level that minimizes the average cost funtion. What is the miimum average cost? (Use Derivatives)

THANKS,

K

 

[skeeter]

using technology (a TI83), a quadratic model of the given data (rounded to a reasonable number of figures) is

y = .006x² + .122x + 102.2

average cost = (total cost)/(number of dozens) = y/x = .006x + .122 + 102.2/x

to minimize average cost, find the derivative of the average cost function, set it equal to 0, and find the value of x that minimizes the average cost ...

y_avg = .006x + .122 + 102.2/x

y'_avg = .006 - 102.2/x²

.006 - 102.2/x² = 0
.006 = 102.2/x²
x² = 102.2/.006
x = approx 130 dozen for the minimum average cost

 

[kjones]

Skeeter,

Can you help me more...

I have the TI-83 and not sure how you came to y= .006x^2 + .122x + 102.2

???

K

 

[skeeter]

you need to enter the data into two lists in your TI83 ... then calculate a quadratic regression based on the data in those lists. if you do not know what I'm speaking of, then reference your owner's manual to find out how to calculate such a regression.

 

[kjones]

Skeeter,

the owner's manual is a bear for those non-math thinking human beings.

K

 

[stapel]

the owner's manual is a bear for those non-math thinking human beings.

The manuals are also the size of small books, which would explain why they cannot easily be reproduced here. If you are needing to be walked, step-by-step, through the buttons to push, then you might want to conference with your instructor or hire a tutor. That way, you will have somebody who can walk you through the various different processes until you learn them.

It isn't hard; just give your instructor/tutor some time to show you the steps a few times. Take notes, and then follow their instructions on a few exercises. With luck, you'll be up to speed in less than an hour. Have fun!

Eliz.

 

[kjones]

Response to Skeeter

You helped me with this problem, but suggested I research the quadratic regression capability on the TI-83. I've done that and can say, 24 hours later, I'm totally comfortable with understanding how that is solved using technology and the assistance you gave me.

Thank you.
K