Additional Math Pages & Resources

Tuesday, June 29, 2010

Larger Doses of Heavy Metal

Continuing from yesterday, today we are weighing a copper nugget. I forgot to tell you yesterday that I bought this huge hunk of so-called COPPER on eBay, from a guy in Arizona.

Here it is next to a prefab fireplace log, to show you its size.

Really, I did. That's before my wife closed my eBay account ...  :-(

The seller delivered it to my front porch, for free, in case you were wondering what the shipping cost! That was before HIS wife closed HIS eBay account ...

We put on gloves for safety, slid the scale underneath, and learned the nugget weighs 151.5 lbs, including 2 small wood planks put there to keep it from ruining our scale. Together the wood weighed 0.5 lbs, so the nugget itself is roughly 151 lbs.



Now how can we tell if it is copper? We can take its volume, multiply the volume by the relative density of COPPER (8.9) and compare the result with the weight from the scale. If this is GOLD (density 19.3) painted to imitate copper, it will weigh more. If it's STEEL (density 7.7) painted to imitate copper, it will weigh less.

OK, how do we find the volume? We can put the nugget into a tub of water, fill up the tub, and measure how much water is in the tub. Then we take the nugget out and fill the tub up the rest of the way. The extra water will give us the volume of the nugget.

One small problem. We can't lift the nugget more than an inch or two off the ground, and we don't have a hoist or handles. But we found a trash container it would fit into (mostly) so we slid the nugget into the trash can.

We turned it up on end and filled the container with water. The container held 8.25 gallons when filled to the brim. But there is a little bit of the nugget sticking up from the top - a prism shape that's a triangle 7 x 4 x 6 by 5 inches long. (Here comes some math...)


We calculated the volume of this piece (4 x 6 x 5 ÷ 2 = 60 cubic inches) and compared that volume to the number of cubic inches in a gallon. A gallon is 231 cubic inches. So we have about a quarter gallon of copper sticking up in the air. If it was below the water line, we would have needed only 8 gallons to fill the trash container. (Did you follow that reasoning?)

We had to dump out the water (since we couldn't lift out the copper). What a mess. But it's only water.


How do we know how much water fits in the trash container? My wife refills it while I start on the math. She says the container holds 10 gallons. So 10 gallons of water - 8 gallons of water = 2 gallons of copper. We had two gallons of copper in the trash container (adjusting for the part sticking up).

How can we confirm that this is correct? 


First we do an estimate. The weight of two gallons of water is (8.35 x 2 = 16.70 lbs or roughly 17 lbs). We know the density of copper is 8.9 times that of water. Therefore 8.9 x 16.7 = 149 lbs.

Eureka, we've found COPPER! Gold would weigh 300 lbs and steel would weigh 129 lbs.

Now let's do it the hard way to confirm our answer. Yesterday we learned copper weighs 8.9 grams per cubic centimeter. A minute ago we learned a gallon is 231 cubic inches, or 3.8 liters.

(The older automotive types in the crowd will know 
that's also the displacement of the Buick V6 Turbo engine, 
later known as the 3.8L engine. See the photo below.)

3.8 liters is 3800 cubic centimeters, because a liter is 1000 cubic centimeters.

And 3800 x 8.9 grams = 33820 grams (the weight of our nugget) ÷ 454 (grams in a pound) = 74.5 lbs. What?! That's wrong. It's way way off! What happened?

Yes. The answer is off, because we had TWO GALLONS of copper, not one!

That's 149 lbs. We got it right.

Isn't this math stuff fun?

(well Doc, I was lifting this huge copper thing when I felt a twinge ...)