There is a new building going up in my neighborhood. It is so tall, that you can see it from the school I teach at. It makes me curious... how far do you think you could see from the top? As my students are learning about trigonometric ratios, I thought this might be an opportunity to bring the community into the classroom. I asked students to make a prediction about how far they could see given the pictures below. I also let them know that if you stand on the beach, you can see the horizon about 4.8 km away for an average height person (a lot of their initial guesses were even less than this). The first step in satisfiying my curiosity is to collect some data. On my bike ride home, I stopped a couple times to take a picture of the building and to use my Invicta MK1 clinometer to measure the angle of elevation to the top of the building. The first stop was just in front of the school and the clinometer measured about 5 degrees. About a kilometer further down the road, I stopped again for some more pictures and measurements. This time, the angle of elevation to the top of the building was about 15 degrees. I used Google Earth to measure a more precise distance between my two stops. It said that the places where I took the measurement were 1097.98 m apart... lets just call it 1098 m. Now to set up a picture of the situation and include some data. And now for some calculations... At this point, I paused to let the class know that I already knew the height of this building... and my answer was not nearly as accurate as I'd like it to be. I was able to find the building permit online and it says the the building is precisely 103327 cm tall... lets just say 103.3 m. So how did I get so far off? I asked my students to brainstorm some possible sources of error. One is my clinometer. I can only read the angle at an accuracy of about +/- 1 degree... and 1 degree of error at that distance makes quite a big difference in the height. Another source of error is the ground... unlike almost every textbook question, my city is not perfectly flat. Looking at a topological map, I can see that the ground rises about 20 meters over the distance that I took measurements. Not enough to look like you're on a hill but enough to make a difference in my calculations. One final source of error might be that my two measurements were not perfectly in line with the building. I'm not sure how much of a difference this might make but I know it added an additional bit of error. All in all, it was a good conversation about how real life is often not as simple as it looks in the textbook. So back to our question... how far can you see from the top? Well, that is where another right triangle can help. Given the height of the building (103.3 m), the location of the building on a hill (60 m above sea level) plus my height of eye (1.74 m), we can find out how far about sea level the viewpoint is. Since this building has a clear view of the ocean horizon, we can calculate the distance fairly accurately (I figured this part out after listening to an episode of the A Problem Squared podcast). So all these number and calculations tell me that, given the radius of the Earth (about 6371 km) and the height of the view (165.04 m) we have a view distance of 45.9 km. There was much rejoicing by the student who predicted the closest.
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