Okay, i have to work out the a scale for a reel of lables?

Question

I know that at a radius of 11.5 cm there are 8000 lables on this reel, and i know at 4.5 cm there are zero lables (the reel hub radius). My mind had drawn blank on working out a scale or equation for working out the intermediate points. (basically i need a way to simply measure the remain radius and know how much is left.

Answer 1

Let R be the radius.
Let N be the number of labels.
N varies with R^2
and (R,N) : (4.5,0) and (11.5,8000) are two points that need to be on a curve of the form:

N=aR^2+b

so let's substitute the two values of R and N that we know to get:

0=a(4.5)^2+b
and
8000 = a(11.5)^2 +b

Solving for a and b I get

a = 71.42 and b = -1446.4
so the formula is

N=71.42R^2-1446.4