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.
"Kama ilivyoelezwa katika makala ya ""Mipaka ya Himalaya"" ya mwaka wa 2015, kilele cha mlima huo ni kilele cha mlima wa Himalaya, Mt. Mlima Everest. Kama unasimama katika India kwenye Ghanga Plain, kwenye milima ya Himalaya, wewe ni amesimama juu ya 180m juu ya usawa wa bahari. Ni umbali wa kilomita 195 tu kutoka mahali tunapoishi hadi kilele cha Everest, zaidi ya mita 8,848 juu ya usawa wa bahari. Ikiwa ungeweza kuunganisha waya wa reli ya reli kutoka Ganges Plain hadi Everest, ungeweza kuunganisha waya wa reli kwa pembe gani? Chagua kati ya 3 °, 5 °, 10 °, 20 °, au 30 °. Kwanza, hakikisha kubadilisha umbali kutoka kilomita kwa mita (au maili kwa miguu) ili uwe unafanya kazi katika vitengo sawa. Umbali wa ramani na tofauti ya urefu hufanyiza pande mbili za pembetatu yenye umbo la mstatili. Unaweza tu kuchora kwa kiwango, na kupima pembe, au unaweza kuhesabu pembe trigonometrically (tan a = tofauti ya urefu / umbali wa usawa). Angle ya chini ya 30 ° "Kutoka kwa ""Mfano wa Kuangalia"" na ""Mfano wa Kuangalia"" katika ukurasa wa SERP, unaweza kupata maelezo ya kina kuhusu jinsi ya kuunda na kuunda safu ya safu ya safu ya safu ya safu." "Kutoka kwa kitabu cha ""Programming Pearls"" (ISBN 02076588)" Kitabu hiki ni kwa ajili ya wanafunzi wa sayansi ya kompyuta na walimu, lakini sehemu hii inazungumza vizuri sana nyuma ya hesabu ya bahasha kwa ujumla.