Math 300 Viewing Tubes

Transcription

1 Math 300 Viewing Tubes Joseph Alameda, Whitney Chadwick, Scarlitte Ponce, and Maria Resendiz August 31, 2015 A description of the data The first step toward completing our task was to collect data. In order to gather our data, we decided to construct two different tubes that were similar in shape but different in length. We constructed two tubes that are similar in shape, cylinders, because this will allow us to make a better comparison give that both have the same structure. The differences between the tubes are the size of the diameter and the length of the tubes. By constructing them different on length and diameter we are able to compare how size affects our field of vision. Our next step was to actually collect our data. We agreed on taking different distances away from our target to determine how distance affects our field of vision. We then recorded the folling data with both using the smallest and longest tube. Figure 1: We are making a formula for the field of view. 1

2 A chart of our data To make sure all of our data worked we converted every length into centimeters. It is necessary to convert the units of the tube into the same units as the distance because both variables are used in the formula to to compute field of view. The proof of this formula is shown in a later section. Yellow Tube(ong) Red Tube(Short) ength cm cm Diameter 5 cm 4cm Table 1: Tubes. Distance Yellow Tube (ong) Red Tube(Short) ength of vision Field of vision ength of vi- Field of Vi- on wall (circle) sion on wall sion (circle) 56.5 cm (Vent 1) 4 cm cm 2 10 cm cm cm (Vent 2) 14 cm cm cm cm cm 20.5 cm cm cm cm cm 27.9 cm cm 2 36 cm cm 2 Table 2: Tube and distance variation. Graph of our data (a) Viewing tubes experiment. (b) Tube lengths. Figure 2 2

3 (a) Visual of field of vision with shorter tube. (b) Visual of field of vision with longer tube. Figure 3 We used two different tubes in our experiment and then varied the distance from the board to collect data. We would measure out a certain distance, check both tubes and then move to another distance. The point we were looking at was fixed. Point of View Formula To calculate the viewing parameter use the formula, π(lr 2 + 2dlr 2 + dr 2 ) l 2. l is the length of the cylinder, r is the radius of the cylinder, and d is the distance from the cylinder to point of view. Proof of Formula Proof. Proof of Field of View Formula. We first assume that the tube is perpendicular to the wall to prevent any error. We then use the axiom that says, between any distinct points there is a distinct straight line. Using the center of both sides of the cylinder we can find the length. With this length we create a ratio between the length of the tube and the length of the radius. We need to find the radius of the field of view. To do this we must add a new r variable that tells us the distance between the end of the tube and the center of the field of view. Using what we know, the length of the tube and D the length between the wall and tube, we let + D be the total distance. To get r we set up a ratio again. = +D r x x = r( + D) x = r + rd x = r+rd To calculate the field of view, we use the formula for the area of a circle, A = πr 2. We replace r with x = r+rd, which gives us A = π(r2 +2Dr 2 +Dr 2 ) 2. 3

4 Applied Question If you had a tube with diameter 2 inches and length 14 inches, how far would you need to stand from the wall to view a 3x4 foot painting on the wall? Is it a problem making an assumption on where the painting is hanging? First we must convert the 3x4 foot painting into inches. This gives us a 36x48 inch painting. Since are field of view is a circle, we then must find out how big of a circle fits this painting. Figure 4: Field of view fitting the painting From the figure we see the diagonal of the painting gives us the diameter of the field of view, which is 60 inches. So the area of the field of view is 900π in 2. Now setting up our formula we get 900π = π(12(1)2 +2(12)(1) 2 D+D(1) 2 ), which gives us 12 2 D = inches or D = feet. The problem does rely of the assumption that the painting was perpendicular to the tube. If it was not perpendicular we would have to take the angle into account and our field of view would be an ellipse not a circle. Thus, our formula only works if this assumption is met. Final Analysis After creating two different sized tubes, our group predicted that the yellow tube which had a larger diameter and length would be less effective than the red tube which was smaller in diameter and length. We then looked at two cases: Will a shorter tube with a smaller diameter yield the biggest field of view or will a longer tube with a larger diameter yield a bigger field of view? After measuring several distances and marking where our viewing area ended on the board, our prediction was realized. The smaller tube with a smaller diameter had a larger field of vision than the larger tube, and after each increase in distance, the gap between the two tubes viewing areas would also increase. Our reasoning was that the larger length of the yellow tube would place it closer to the board thus reducing the area that could be seen when someone looked through it. After taking several measurements and placing our data into a table we were able to compare our prediction to the actual results and concluded 4

5 that the smaller tube with a smaller diameter yielded a better field of vision. This analysis of course relied on the assumption that our view was perpendicular to the board, the diameter was fixed and that are measurements were exact. 5