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So, about a decade ago my company came up with pricing for some banners that we sell. the prices are as follows.

$43.68 for a 3x4 banner
$44.52 for a 3x6 banner
$46.36 for a 3x8 banner
$50.00 for a 3x10 banner
$52.54 for a 3x12 banner

and I can not figure out where these prices came from. The guy who wrote them up quit before I started, and I need to figure out the equation to extend the pricing up and down.
Here's what I DO know.
The equation is based off two things
The cost of the banner per square foot
The cost of labor
I do not need to figure out the factors that went into pricing for either, I just need to know what numbers they are. Best guess for labor was 63 dollars, it might not be, but if that works, it sounds good to me.
my attempt was to figure it out using substitution with a system of equations.

12(sqft) * X($/sqft) + 63($/hour) * Y (hours) = 43.68 and
18x + 63y = 44.52

with a second set of

24x + 63y = 46.36 and
30x + 63y = 50.00

BUT the first set gives me 

x=0.14
y=0.66667

and the second set gives me

x=0.606667
y=0.504762

which leads me to believe that the hours per banner change. Meaning the y in each equation is different. Is there a way to determine what these two variables are, even though one changes, probably linearly? If not, I'll just do a whole new equation, the only issue is the number of variables going into each of these variables. Thanks.

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    $\begingroup$ I think you're drastically overthinking this. Why on earth is it important that the new prices be mathematically related to the old prices? Just work out newly, without reference to the old prices, roughly what the cost is to produce a banner measured in terms of material and labor, add in a profit margin, and sanity-check your result against the old prices. (Note: Yes, I've actually worked in a print shop.) $\endgroup$ – Wildcard Jan 3 '17 at 23:55
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    $\begingroup$ I don't know, all I know is the CFO of the company (not a print shop, a beverage distributor with a print shop in it) dropped this in my lap about 20 minutes ago and I'm freaking out... I can do that, I was just worried about ink coverage and laminate wastage and all that, and was hoping to just refer to the old equation $\endgroup$ – Sharkn8do Jan 3 '17 at 23:58
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    $\begingroup$ Right, but you have no evidence at all that the old equation properly accounted for the production overhead, laminate wastage, etc. (The print shop I worked in was an internal shop as well, incidentally.) The only usable equation would be one that showed the calculations of costs. Why do you imagine that ink prices haven't changed at all in the last decade? $\endgroup$ – Wildcard Jan 4 '17 at 0:01
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I plotted the cost as a function of the area of each banner here. I also made two regressions--one linear and the other quadratic. I recommend you use the quadratic regression to project the prices, because the quadratic more closely matches the data trend and any extrapolation using this regression will give a higher price than the linear regression (you are less likely to undercharge).

plot of data

The quadratic regression gives the equation

$$y=0.010317x^2-0.10857x+43.34$$

Where $y$ is the cost and $x$ is the area of the banner.

In the end, I agree with Wildcard's comments; you should use a heuristic method to find the best price for your product, but this is a good starting point.

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Each time you go up a size you add 2 feet to the length. The added cost for these 2 feet varies from 0.84 to 3.64, which is quite a variation. This shows you will not be able to generate a formula of $A + B(length)$ that fits the old data, as $B/2$ should be the added cost per square foot. Clearly labor is not $63$ as all the prices are less than that. You can do a least squares fit, shown below, which gives price =$38.14+1.16length$

enter image description here

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  • $\begingroup$ Things take less than an hour. The belief is 30-40 minutes $\endgroup$ – Sharkn8do Jan 4 '17 at 4:15
  • $\begingroup$ You didn't say 63 dollars per hour. Do you think the labor is the same regardless of the size of the banner? Any variation can be part of $B$. $\endgroup$ – Ross Millikan Jan 4 '17 at 4:17
  • $\begingroup$ I think there is a possibility for variance, but why it would sometimes be below the line and sometimes be above the line is beyond me $\endgroup$ – Sharkn8do Jan 10 '17 at 17:22
  • $\begingroup$ That is always what happens on a least squares fit. We adjust the parameters on the line to be as close to the data as possible. That leads to some points being above and some being below. You can reduce the variance by adding more parameters, as pseudoeuclidean's quadratic fit shows, but the points are still some above and some below the curve. $\endgroup$ – Ross Millikan Jan 10 '17 at 17:39
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If you use the formula .4*area + 38, you'll get approximately what the current prices are. It's an easy formula to apply if you want the new prices to seem consistent with the existing prices. However, if your new sizes include dramatically larger areas, then spend some time investigating costs.

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