Creating a function based on data using integration

[JFonseka]

Homework Statement

I first posted this in the pre-calculus section, then the guy who first came up with the formula hinted at integration, so I guess it's meant to be in the calculus section

Now this isn't a homework question, it's something me and some others are looking into, and someone posted this function, and I'm not sure how he worked it out:

Creating a function for Elvis' album sales based on the RIAA certifications of his albums that have been certified so far

69 albms - over 500,000 sales certified
39 albums - over 1,000,000
19 albums - over 2,000,000
11 albums - over 3,000,000
4 albums - over 4,000,000 and 5,000,000
2 albums - over 6,000,000
1 albums - over 9,000,000

Let exclude the 4 albums over 4 million, which could be estimated on their own and disturb the linear repartition of figures.

I set X = 500000*x

Then Elvis statistics are almost perfectly equal to the function 1/(1,23*x + 0,278).

Let x = 1 if you want to find albums that sold over 500,000
Let x = 2 if you want to find albums that sold over 1 million
etc.

So what he's done is used the figures from the albums certified to create an equation. So therefore if you substitute 1 for x in the function, which is actually equal to 500,000 you will get all the albums that certified for over 500,000 sales, it's not exact, but it's a close enough function, how did he work that function out?

Homework Equations

None

The Attempt at a Solution

I have no idea!

 

[Dick]

Probably just drew a graph and concentrated on the points that seemed to fit a smooth curve. Adjusted a and b in f(x)=1/(ax+b) to fit it. You can use least squares type techniques to help to you estimate the constants. But I'm confused. I put x=1 into 1/(1.23*x+0.278) and I get 0.663. That's not 67. Do you mean I should multiply by 100 also?

 

[JFonseka]

Probably just drew a graph and concentrated on the points that seemed to fit a smooth curve. Adjusted a and b in f(x)=1/(ax+b) to fit it. You can use least squares type techniques to help to you estimate the constants. But I'm confused. I put x=1 into 1/(1.23*x+0.278) and I get 0.663. That's not 67. Do you mean I should multiply by 100 also?

Yea you have to multiply by 100, I got 66.3 too, but I guess he rounded up, not sure why he rounded up, but the other values seems to indicate that's what he did.

 

[Dick]

Yea you have to multiply by 100, I got 66.3 too, but I guess he rounded up, not sure why he rounded up, but the other values seems to indicate that's what he did.

Like I said, you can use least squares to fit almost any smooth curve pretty well. Especially if you start throwing out points that don't fit. And use creative rounding to boot.

 

[JFonseka]

I'm not too sure what that is, but I'll look at it now!

Thanks Dick.