Thursday, January 14, 2010

SkyCity Restaurant Diameter (Fermi Problem)

These last couple of weeks since Christmas have been very busy for my wife and I. We are preparing to move to a new apartment in Boulder, we traveled locally to a family get together for Christmas and for New Years, and we have just returned from a trip to Seattle.

After all this running around frantically (and having little time for blogging btw) I decided it would be nice to blog about a nice dinner I had with my wife at the SkyCity restaurant atop the SpaceNeedle!




We were within a short walking distance of the SpaceNeedle, and so 47 min before our reservation we headed out.
This being Seattle in January, the weather was drizzly and cool, with fog. Surprisingly, even though it was 47F, the air felt warmer than an equivalent temperature would in Colorado. Any rain that came down did not feel like the soaking-freezing kind that is common in Colorado, but just seemed like an ever present misting. My wife and I rather enjoy such weather, so we were happy on our walk.

On our ride up the elevator we are regaled by the busboy about how tall the SpaceNeedle is, how fast the elevators go and what sort of earthquakes and winds the tower can withstand.

As we looked out the window of the elevator, we were a little disappointed that the view would be obscured by the rain and fog. The busboy commented: "The view from the SpaceNeedle, on a clear day is unrivaled, with views of the sound and Mt. Rainer. The SkyCity reasteruant rotates once in exactly 47 min, and is powered by a 1.5 HP motor. So you should be able to go around at least once or twice during your meal!"

We were shown to our table and our server (who earned her tip!) gave us our menu and took our order. (I wont go into detail, but it was not cheap!).
As my wife and I sipped our martinis, we gazed out of the window at the view, which despite the fog and snizzle was still really nice!

As we awaited our food, I wondered out loud about the diameter of the restaurant. When our server returned, with our food, I asked her about the diameter.
"Wow, no one has asked that one before! They always ask how high the SpaceNeedle is or some other question. I am not sure! I will ask the others and see if I can find out." And off she goes to ask around on the diameter of the SkyCity restaurant.

"Well heck..." I say to my wife; "I have enough information right now to estimate the diameter.

"Why not look it up online on your iPhone?" My wife asks as I look around for a piece of paper to scribble on.

"What is the fun in that?" I reply; "I will do that later for verification."

So, on to one method of estimating the diameter of the SkyCity restaurant!

------------------ Start Problem --------------------

So, the restaurant rotates once every 47 min. The dining seats and table are on a rail while the windows and the rest of the super structure remain stationary.
This can be seen in the cutaway drawing below.


As you can see in the picture, the windows slope outward on the rise. This means that there is an average diameter, as the max diameter is near the ceiling, and the min are at the floor. I decided to say the diameter of would be at my eye level as I was seated. Since this was a back of the envelope type problem, I did not really expect to estimate much better than a few percent anyhow.

So, first lets draw a rough sketch:



By simple trigonometry, the distance S is given by:
 
Where theta is in radians.

For a quick and dirty estimate, I would assume that the width of the window we were sitting in front of represented a short enough chord to use a small angle approximation for the radius.

The small angle approximation states that when the angle as measured in radians is much less than 1, or "small", then Sin[angle] ~ angle. Using this approximation and rearranging terms, I get an estimate for the radius R.

Then by adding up all these little chords I can approximate (and slightly underestimate) the radius of SkyCity.

Now lets consider some numbers:

The window pane I  estimated to be about 6 feet in length. There is some error as I did not have a measuring tape, and my wife insisted I not get up to measure with my arms.
Taking out my handy dandy iPhone I timed the transit of the window to be approximately 57.5 seconds.
This allows me to set up the following ratio that is equivalent to adding up the chords:




On solving for R I estimated the radius to be:

R = 93.7 feet.

Done.

------------------------ End Problem ----------------------

A few minutes later our server returns:
"You know, no one knows the answer. Sorry about that."

"That's ok. It is about 94 feet in diameter, give or take." I reply.

"Really? Where did you find that out?"

"I estimated it." I reply.

My wife and I return to gazing out the windows and at each other while enjoying our meal. For desert we ordered the Lunar Orbiter, which was wonderful and foggy! As we finished off the last of the ice-cream, I decided to look up the diameter on my handy dandy iPhone using some google fu.
According to the SpaceNeedle  web site the diameter is 94.5 feet. This is surprisingly close to my estimate.

In fact, my estimate is within 0.9%, or around 1.25 feet of the real answer. That is simply too fortuitous. Perhaps a future blog will deal with error propagation and error sources in such an estimate.

Well, all in all it was a wonderful evening! I was with my wonderful wife, at a nice restaurant, and I got to play with a Fermi problem.
=)


Images from:
http://bibliodyssey.blogspot.com/2007/03/century-21-exposition.html