Back-of-the-Envelope Calculations: Height of the Himalayas
Let's think about the highest peak in the world, Mt. Everest. If we stand in India on the Ganges Plain at the foothills of the Himalayas, we are standing at about 180 m (590') above sea level. It's only 195 km (121 miles) map distance from where we're standing to the summit of Everest at over 8,848 m (29,028') above sea level. If we were to stretch a nice straight wire for a cable car from the Ganges Plain to the summit of Everest, at what angle would the cable rise? Choose from 3°, 5°, 10°, 20°, or 30°.
First, be sure to convert the distances from kilometers to meters (or miles to feet) so that you're working in the same units. The map distance and the difference in elevation form two sides of a right triangle. You can simply draw it to scale, and measure the angle, or you can calculate the angle trigonometrically (tan a = elevation difference/horizontal distance). The angle is less than 30 degrees.
References and Resources
This SERC page describes the use of Back of the Envelope Calculations
A View from the Back of the Envelope: 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.