Additional Math Pages & Resources

Monday, October 12, 2009

You can never be too tall or too thin or too rich

But a wall can be too tall and a coat of paint can be too thin.

I need to paint a tall, strangely-shaped wall in our house. The painters left me a little bit from the last job, so I can have the color matched down at the paint store. The question is - how much do I need? The color is sort of a medium blue. I want to cover it with white. I know I will need more than one coat of paint.

The wall is a complex shape. Here's a drawing I made of it.

We can calculate the area if we think of it as a rectangle at the bottom with a right triangle on top.

The rectangle is 172 inches long by 93 inches high. That's 16,000 square inches.

The right triangle is about 150 inches on one side, and 48 inches on the shortest side, so that's 7200/2 or 3600 square inches.

I round the totals to 20,000 square inches. Divide by 144 to get 139 square feet.

A quart of paint will cover 400 square feet. I could apply at least 2 coats of paint if I buy a quart.

So I bought the quart of paint and put the whole can of paint on the wall (it did 2 coats plus a bit more). As I sat there looking at my wall, I wondered, How thick is the paint?

My wife said, Excuse me? How thick is the paint?

Yes, I said. Putting a quart of paint on the wall should be easy enough to figure - how thick is the coat of paint?

Sounds like a Blog Problem, she said, turning back to her mystery novel...

So here we go. First we need a few rules:
  • Let's make an assumption that 10% of the paint gets wasted by the brush, spilled, etc. 
  • We'll ignore "soaking in" and assume the paint dries on the surface.
  • It will be easier to do this in metric units. Let's convert to metric before we begin.
A square foot is 930 square centimeters. Multiply 139 square feet x 930 to get about 130,000 sq cm.

A quart is 946 cubic centimeters. If we waste about 10% we'll have 850 cc of paint.

Our question can now be stated: How thick a layer will 850 cubic cm of paint create if poured out on 130,000 sq cm of wall?

If the wall was a floor we would just pour the paint out, let it spread, and let it dry. But here's another way to think of it.  Let's re-shape our 850 cc cube to 20 cm x 20 cm (400 sq cm) x 2.125 cm thick (20x20x2.125=850).

Imagine the paint as a block of cheese. We slice the block, and simply spread the 20x20 cm slices around on our 130,000 square cm piece of bread. How many slices do we need?

If we divide 130,000 (wall) by 400 (slice) we get 325. We need 325 slices of the paint to cover the surface of the wall. So we need to slice the 2.125 cm thick block into 325 slices. Each slice of cheese/paint is 1/325th of 2.125 cm. That's .0065 cm, or if converted to mm it's .065 mm.

Seems thin, doesn't it? Let's recheck our calculations.

.0065 x 130,000 = 845  ≈ 850 It checks!

Still a bit uncertain, I found a company that makes ultra-sonic paint testing gauges and discovered  most painters put one coat of primer and two coats of finish paint on interior walls. Here's how the measurement readings look using their ultra-sonic paint thickness tester:

These readings are shown in mils but can be converted to mm.
The three coats of paint vary from 1.4-4.1 mils of thickness. If we take my paint thickness of .065 and multiply by 39 to convert to mils, we get 2.5. That's close enough for me.

These cool gauges are $500-2500 and another trip to the paint store would have taken at least as long as these calculations. I saved both money and time.

See how useful math is?! I exclaimed (standing tall) with pride.   Zzzzzzzzzzzzzzz ... she replied.

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