Painter's problem

A painter has varying paints which vary in opacity, x. He wants to figure out how many coats of paint, y, are required to make a certain paint of opacity x appear at full opacity. What equation should he use?

Lots of paint, drop-cloth & a big-a$$ brush. :D:D:D

Assuming full opacity is 100, and x is a percentage (such as 0.25),

y = 100/x


Or am I missing something here?

It's additive.

Say he has 50% opaque paint. Two coats and this becomes 75% opaque.

In each case, you'd have to take the sum of the geometric series. In our 50% example,



1 = x(1 + 1/2 + 1/4 + ...)



Sum of a geometric series is

S = a(1-rⁿ)/(1-r)


In our example, a = x, r = 1/2 and S = 1

1 = x(1-(.5)ⁿ)/.5

And I'm too dumb to go on from here :cry: , but hope this leads you somewhere!

See? My my posts are of varying bull**** x. :D

x has to be of full opacity, and then we have y=1.

That's what I was confused about.

If x < 1, then y = infinity.

Quote:

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Originally posted by mendhak
That's what I was confused about.

If x < 1, then y = infinity.

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we cannot solve the equation (1-x)^y=0 for x<1

This is like the half-distance paradox.

Me and Cameron Diaz are standing a few meters apart in a room. Between us, there are two halves. If I move half forward, then there's another two halves between us. Like this, there are an infinite number of halves.

Of course until we start making out, but that doesn't figure into mathematical terms. :)

Quote:

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...appear at full opacity...

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Which, for this sake, we'll say is >98% opaque.

then we have
(1-x)^y=0.02
y=log_1-x 0.02

Quote:

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Originally posted by kedaman
then we have
(1-x)^y=0.02
y=log_1-x 0.02

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Hmm.

y=log_1-x 0.02

gives me a negative result when x is .5.

then your calculator is broken, mine gives ~5.64