High Tech High has a flagpole in the front of their school. The principle needs to buy a new flag, but he needs to know the height of the flagpole so the flag and the pole will look proportional. No one knows the height of the pole so we need to use multiple methods to find the height of the flagpole.
Process & solution
We first started this problem with guessing what we thought the height was. My initial guess was about 35 feet. I thought this because I thought of how many times me's could fit withing the flagpole. I'm about 6 feet, so I got me around 5-6 times, which amounts to around 35 feet. After our initial guess we had to learn similarly. Each of our methods involved similarity, so it was a big thing we had to learn. Mr. Carter gave a definition that described similarity perfectly, "Same shape, different size". That is exactly what similarity is, but first we had to learn how to prove that the two shapes were the same.
The shadow method
This theorem applied to the shadow method because we had two corresponding angles that won't change. Since the sun is in a fixed position, the angle of the sun to the end of the shadow will be the same on the person and the object that you are trying to measure. As well as the angle from the base of the object to the end of the shadow, which is 90 degrees. These two angles are all we need to prove the AA Theorem. Proving that triangles are similar allows us to create a proportion to find the height of the object. In this case the object was the flagpole. My table and I came up with the proportion of persons height/persons shadow length = pole height/ pole shadow length. We began to measure each of my group mates height and then their shadow length, after doing this we found the average between us, 66 in. / 106 in. and used that as our control height. For the shadow method, we discovered that this worked because of the Angle-Angle Theorem. The AA theorem states that if two angles that are corresponding within a triangle are similar, the third and final angle will be the same and create the same shape of the first triangle.
We then measured the length of the flagpoles shadow, it was 480 inches. Since we did not know the height of the pole, we gave it the value of x . After setting up the proportion we crossed multiplied to solve for x. Using the shadow method, we came up with the value of 25 ft for the height of the flagpole.
Mirror method
We were able to use the Angle-Angle Theorem for the mirror method aswell. When you look down into a mirror, the angle you are looking at it from will be the same angle that is reflecting from the object you are looking at, creating our first similar angle. Our next similar angle is the 90 degree angle formed from the base of the object and person to the mirror, creating our second angle. We now have enough angles to prove similarity. We created a diagram and inserted different variables for each length. After creating the proportion with our variables, we practiced with various objects around the school. After some practice, we went to the flagpole to use its measurements in our proportion. Our first variable was a persons height, 61 in, our second variable was the length between the person to the mirror, 35 in, and our third variable was the distance from the object to the mirror, 137 in, our final variable was the object height but that was what we were solving for. After inserting the measurements in their proper variables, we crossed multiplied to solve for the objects height. Using the Mirror Method, we concluded that the flagpoles height was 20 ft.
clinometer method
When we used the clinometer method, we only used one triangle so there was no need to prove similarity. A clinometer is a tool of measurement that uses a straw, a protractor and some string to find the height of an object. This method required us to form an isosceles triangle with our bodies and the object. Forming this type of triangle was crucial because in an isosceles triangle, the base length is the same as the height. All we had to do was form a 45 degree angle that also allowed us to view the flagpole. After accomplishing this, we measured the distance from our eyes to our feet, then from our feet to the base of the object. We then added these two measurements together and knowing that in an isosceles triangle two sides are the same length, the two measurements were the same and we knew what the height of the object was. Which was 25 ft.
Height of the flagpole
Using the three different methods to find the height of the flagpole, the average is 23.3 ft. This is the most accurate measurement because if we take the average of the three different methods we used, we could find the in between. The answer- based on our calculations and measurements - should be around 23.3 feet.
problem evaluation
This problem was a lot of fun. What kept me interested most was seeing all the different ways we could use to find the unknown height of an object. We learned a lot about angles and I think that is most useful because in our upcoming years we will be learning a lot about geometry, so getting a headstart on it will be very useful.
self evaluation
If I were to give myself a grade throughout this whole unit, I would give myself an B. There is a lot of things I wish I had done differently throughout this unit, for example, at first I didn't understand a lot of things and I should've taken that as an opportunity to go to tutoring, but I didn't. A big change I need to make is to go to tutoring if I need it, and this unit has helped me realize that.