Estimating Total Area: Help with Calculating Missing Space on a Flat Plat

[higha level]

This is not a HW but someone said in a previous thread that it sounds like it so I will post it here because it fits the format I guess.

Homework Statement

I have a flat plat that is x by y. It has an unknown number of holes in it but I do know the diameter of the holes is D.

I also know that s/D = 4 where s is the spacing between holes of diameter D. Using this how can I estimate the total area absent in the flat plat, i.e. the amount missing.

Homework Equations

s/D = 4

area rectangle = x*y

area circle = pi*r^2

D = 0.05"

x = 4

y = 3

The Attempt at a Solution

I've been over thinking this but I can maybe estimate the amount of circles of diameter D can possible fit in there and leave it at that but is there a mathematical way to do this as oppose to me just drawing it out and guessing the amount of holes.

 

[sezw]

find s find the total area of x and y d=.05" d/2 = radius of the wholes and find the area of one and divide the area of x and y by the area of the holes and find the equation this is basic math with a twist to it why did u need help?

 

[higha level]

I don't see where you used s.

Also, I do not know how many holes there are so I can't just find the equation. If I knew the amount of holes and therefore cannot calculate the area of holes.

The area of holes is what I am trying to estimate. I'm thinking I have to use the expression s/D = 4.

It might be something simple but I miss simple things. Sorry if it offends you.

 

[songoku]

Hi higha level

Yes, you have to use the expression s/D = 4 to find s. Actually, you can determine the number of holes that contained in the plat.

Let's consider the x-side. Basically, you have x, D, and s. Just think a simple one. If D = 1, then s = 4, so if x = 3, you can only have 1 circle on x-side. Now do it regarding the data from the question.

 

[higha level]

That's what I was saying I did.

D = 0.05" so I got a s= 0.2".

x = 4 inch so therefore there probably can only be about 15 circles along x. And subsequently 11 in the y=3. This gives an estimate of 165 circles and I can calculate the missing area, etc. I previously did this already and moved forward. The question wasn't a homework problem but someone said it sounded like it.

My question was is there a simple way to calculated it with an equation using s/D to find that missing area without finding the amount of holes there are? Still coming up with a relatively close answer?

Sorry if I didn't ask correctly before.

 

[songoku]

I don't think there's a way to find the missing area without finding the number of circles...

 

[Mentallic]

Just for clarification, is s the distance between the outsides of the circles or the centres of the circles?
Also, I am assuming here that the circles are formed in a pattern such that the first and second rows are in a straight line parallel to each other, except the circles in the second row are not vertically beneath the circles in the first row. As such:

http://img687.imageshack.us/img687/3927/circleareaapprox.png

I was surprised to find that the diameter of each circle isn't needed to find the answer, all you need is x and y.

I'll give the answer now, and if you're interested in the maths behind it, just ask. And I haven't tested the result myself so be weary. Use at own risk

The area, A, of all the circles combined by using the layout I showed above is given by A\approx \frac{\pi}{25\sqrt{3}}xy
The constant would be more appropriately approximated: therefore A\approx 0.073 xy

If you can post a picture of what the layout of the design is, we could give a better approximation.