# Back-of-the-Envelope Calculations: Percentage of Copper in Ore

#### Summary

*Question*

Suppose that you are building a new house. It will take about 90 kg (198 pounds) of copper to do the electrical wiring. In order to get the copper in the first place, someone needs to mine solid rock that contains copper, extract the copper minerals, throw away the waste rock, and smelt the copper minerals to produce copper metal. Rocks mined for copper typically contain only very small percentages of copper—about 0.7% in the case of most of the big porphyry copper deposits of the world. How much rock would someone have to mine in order to extract enough copper to wire your new house?

## Assessment

*Answer*

A block weighing about 12,860 kg (almost 28,300 pounds, or over 14 tons)!! The copper ore forms only 0.7% of the rock. We need to find a number that we can multiply by .007 to get 90 kg. Dividing 90 kg by .007 will give us that number—approximately 12,860 kg. Multiplying that number by 2.2 pounds/kg would give us the number of pounds (about 28,300 pounds). Dividing by 2000 pounds/ton would give us the number of tons (more than 14 tons). Remember that, of the 12,860 kg mined, 12,770 kg were thrown away to get your 90 kg of copper... Now, think about the number of people on your street. How much rock would have had to be mined in order to wire all the houses on your street? How much rock was thrown away?

## References and Resources

This SERC page describes the use of Back of the Envelope Calculations

A View from the Back of the Envelope (more info) : This site has a good number of easy simulations and visualizations of back of the envelope calculations.

The Back of the Envelope : This page outlines one of the essays in the book "Programming Pearls" (ISBN 0-201-65788-0). The book is written for computer science faculty and students, but this portion speaks very well to back of the envelope calculations in general.