This is part 2 of my post on Trigonometry and measuring trees.
I made this post in two parts because a lot of learning has happened this week that would be overwhelming, both to write and to read, in only one post. I left off with the grand task of measuring my Christmas tree. I felt it would be very simple, since I could do the calculations and then measure the tree directly to check them. I had only one problem, I could not find a protractor at home.
You see, the way we measured our school when I was in tenth grade was by standing an x distance from the wall and, using a protractor, we would measure the angle from our the spot we stood to the top of the school. That way we could use the tangent formula tanθ=opposite side/adjacent side, and get:
Tan(angle) x (distance from wall) = height of school
To make up for the lack of protractor I rummaged the App Store to find al alternative. I needed an app that would measure the angle of inclination of the phone as I used it to look at the top of the christmas tree. I settled on Angle Meter.
This app lets me lift the iPhone and measure the angle of inclination. It's purpose is to use the camera as a viewer and to measure the angles of what you are looking at.Their site will explain it better that I just have. The app was very sensitive and it took me a few tries to figure out how to tweak it for my purpose, like subtracting 90 deg from the angle shown.
When I felt ready I took the distance from my position to the tree. Then I measured the angle (making sure I subtracted 90 deg from the angle I got) by placing the iPhone on a small table to keep it steady. I plugged these two measurements into the equation above. And got the height of the tree as seen in the picture. Simple right?
Not so. Even as I went to be thrilled about what I had just done, a clearer mind next morning showed me my mistake. You see, I did not get a perfect result the first time. So I kept at it a few times thinking the problem was the sensitivity of the app measuring the angle. At the distance I was from the tree, a .10deg difference was almost a 12 inch difference.
In my first attempt, the record of which I sadly erased, my measurement came up short...by a lot. That was immensely frustrating. Then once I knew the hight of the tree, I kept measuring to get the right angle. I at least was honest with myself enough as to try to "get" that angle by using the phone app to measure the top of the tree. Therefore once I succeeded, it felt like a true accomplishment. What then did the morning reveal?
The next morning I realized that it was impossible for my calculations to be right since I did not account for the height of the table I had placed my phone on. If you look at the diagram of the figure measuring the school, you will notice that in order to get the true height of the school you will need to ad the height from the floor to the compass. The reason my first measurement was short, could have been because I needed to account for the table's height.
What is important to keep in mind is that I got a "right" result because I knew what that result should be and I did not stop until I got it. The phrase "even a broken watch is right twice a day" comes to mind.
The realization of my mistake was a blow. Even though my efforts at learning Calculus are only for self growth, the fact that I could have given incorrect data of that magnitude, without realizing it, was disturbing. I pledge from now on to double check all my work for follies like these.
To rectify my mistake, I regrouped and tried another approach. Even when the Angle Measure app was very good at what it did, I was using it for a different purpose. I decided to get a different app whose purpose was closer to what I needed. That is how I got to Easy Angle, an app that takes a picture and then let's you measure angles in it. I grabbed my tape measure and looked for something to measure.
I found a column in a parking structure and measured 120 inches from that column with my tape measure. I left the tape measure in place and stepped back to take the picture with the Easy Angle app. Using the tape measure as a guide aI measured the angle. I had to repeat that process a couple of times to get horizontal line exactly right. At least the Easy Angle app kept the photo static so I could work on it over and over. Once I got that horizontal line just right (35.31 deg), I ran the numbers. That column should be 85 inches. And according to my tape measure, that column is about 85" tall.
The bottom line of my adventure is that one should always be aware that knowing what a result should be can influence the process of getting that result.
It is like finding only what one wants or expects to find.I put forth a very situation that sometimes happens to people. A flat tire early in the morning.
When one gets a flat tire at the start of the day, it is normal to think : This is going to be a bad day. And sure enough, most of the time the day is quite horrible. Yet, was the day horrible because it was "destined" to be a bad day? Or did were we just looking for bad things to happened and made every little nuisance a bad thing?
That is the question in life and in math.
I appreciate your feedback so sound off in the comments and let me know how I am doing.
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