(The title is a play on a popular ad copy format)
I saw this article on Google interview questions, and so of course I wanted to estimate the weight of the Empire State building.
I like to do things like this twice. First I just think about it for a second or two and guess. Then I actually estimate. Then I look up the actual answer to see how close the guess and the estimate were.
For the guess, I thought about a million tons, but I didn’t like the idea that the Empire State building might weigh 2 billion pounds, so I backed off and said 100,000 tons.
For the estimate, I took an odd route. I remembered reading some time ago that the Empire State building actually weighs less than the material that was excavated for its construction site. So rather than worrying about how dense the building is, or how it tapers as it goes up (88 floors?), I decided to estimate the excavation and go with that. So:
- Take 50 meters on a side for the excavation.
- And 20 meters deep.
- That’s 50 * 50 * 20 = 50,000 m^3.
- I think Manhattan is granite.
- I think lead/mercury/gold are about 11 times as heavy as water.
- So I take 5x as the multiplier for granite compared to water.
- A cubic meter of water is a metric ton.
- So the excavation would be 50,000 * 5 = 250,000 metric tons.
- Converting and rounding up, that’s about 300,000 short tons.
From the wikipedia page the Empire State building weighs about 370,000 short tons.
I underestimated the area of the building/excavation, and overestimated the density of granite (after underestimating the density of lead, mercury and gold). As often happens in situations like this, the two factors counteracted each other and my estimate was off by just about 20%. I’m particularly pleased that the excavation trick worked out. It simplified the estimation significantly.
And my guess was within a factor of 4x, so I’ll take it as a win as well. I would have been closer sticking with a million tons, or especially with a billion pounds, which was my initial upper bound and the reason I didn’t go with a million tons.