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Ahmed – Davey – Morley
9/25/14
11A
Let’s Have Math Class Outside!
Mathematics is a universal concept that can be applied to everyday real life
situations. This outside math activity demonstrates that. Our team consisted of three
members: Thanvir Ahmed, Megan Davey, and Mitchel Morley. We had to solve four
math problems outside using unique materials including a compass and measuring
tape. All the problems had to do with measuring the given lengths and angles of objects
in order to compute the distance or length of the unknown side or object. Different
methods were performed in order to solve each of these four problems.
The first problem was practice orienting ourselves using the compass. We had to
mark a spot on the ground and then navigate our way to different distances. After
stopping and moving three times, we ended up back where we started, give or take a
few inches. Since the compass was handmade it was not as accurate as it could have
been. A more accurate measuring device would improve our measuring technique, and
would allow us to land exactly at the beginning position.
Figure 1. Diagram of Path Taken
Ahmed – Davey – Morley
9/25/14
11A
The figure above is a diagram of the path taken for Problem 1. Point A was the
starting position. From Point A we traveled a distance of 20 ft. on a bearing of 40° and
reached Point B. From Point B we traveled a distance of 20 ft. on a bearing of 160° and
reached Point C. From Point C we traveled a distance of 20 ft. on a bearing of 280° and
reached the original starting position of Point A. Since all of the sides had the same
lengths, it is clear that the triangle is an equilateral triangle.
The second problem was to find the height of the flagpole. Since we could not
directly measure the flagpole, our method for solving this was to measure the length of
the flagpole’s shadow, and compare it to the length of Mitchel’s shadow. Also, we
compared the flagpoles height to Mitchell’s height. The flagpole and its shadow form a
90 degree angle, same with Mitchel and his shadow. Since both form right triangles, you
can set up a ratio that compares the two triangles, and then cross multiply to solve for
the height of the flagpole. We calculated the flagpole height to be 33.75 ft. the true
height of the flagpole was 35ft. This answer was not off by a lot but, in order to get a
more accurate height we could have used a better measuring device to find the length
of the shadows.
𝑓𝑙𝑎𝑔𝑝𝑜𝑙𝑒 ℎ𝑒𝑖𝑔ℎ𝑡
𝑓𝑙𝑎𝑔𝑝𝑜𝑙𝑒 𝑠ℎ𝑎𝑑𝑜𝑤
=
𝑀𝑖𝑡𝑐ℎ𝑒𝑙 ′ 𝑠 ℎ𝑒𝑖𝑔ℎ𝑡
𝑀𝑖𝑡𝑐ℎ𝑒𝑙 ′ 𝑠 𝑠ℎ𝑎𝑑𝑜𝑤
𝑓𝑙𝑎𝑔𝑝𝑜𝑙𝑒 ℎ𝑒𝑖𝑔ℎ𝑡
6 𝑓𝑡
=
90 𝑓𝑡
16 𝑓𝑡
16(𝑓𝑙𝑎𝑔𝑝𝑜𝑙𝑒 ℎ𝑒𝑖𝑔ℎ𝑡) = 540
𝑓𝑙𝑎𝑔𝑝𝑜𝑙𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 = 33.75 𝑓𝑡
Figure 2. Equation for Flagpole Height
The figure above shows how we used proportions to find the height of the
flagpole.
Ahmed – Davey – Morley
9/25/14
11A
Figure 3. Flagpole Triangle and Mitchel Triangle
The figure above shows the method we used to calculate the height of the
flagpole. Both Mitchel and the Flagpole are perpendicular to the ground. The sun casts
a shadow on both the top of Mitchel’s head and the top of the Flagpole, and because
these measurements were taken at the same time the triangles that are formed are
similar triangles. We then took this fact and used proportions to solve for the height of
the Flagpole as shown in Figure 2.
The third problem was to find the distance between two marked poles. Sounds
easy, right? But, there’s a catch. You have to find this distance without crossing the
street. Our method to find this was to first measure the distance from the first pole to a
spot we marked on the cement. Next, we found the angle from the first marked pole to
the second pole using the compass. Then, we found the angle from the spot on the
cement to the second pole. Since all angles in a triangle have to equal 180 degrees, we
took the two found angles and subtracted them from 180 to find the third angle. We
Ahmed – Davey – Morley
9/25/14
11A
then used the Law of Sines to find the distance between the two marked poles. The
distance we calculated was 57.53 ft. The true distance was 61.2 ft. This could have
been more accurate if we used a better measuring for finding the lengths and angles.
sin 𝐴 𝑠𝑖𝑛𝐵
=
𝑎
𝑏
sin 84
𝑠𝑖𝑛36
=
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑜𝑙𝑒𝑠
34𝑓𝑡
𝑠𝑖𝑛36(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒) = 𝑠𝑖𝑛84(34)
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 =
𝑠𝑖𝑛84(34)
𝑠𝑖𝑛36
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑚𝑎𝑟𝑘𝑒𝑑 𝑝𝑜𝑙𝑒𝑠 = 57.5 𝑓𝑒𝑒𝑡
Figure 4. Distance between Marked Poles
The figure above shows the method used to find the distance between the two
marked poles. We used the Law of Sines to find this distance because the triangle
formed was and Angle - Side – Angle (ASA) triangle. This is shown in Figure 5.
Figure 5. Distance between Poles
Ahmed – Davey – Morley
9/25/14
11A
The figure above shows the method used to find the distance between the two
marked poles. To find the distance between the marked poles we first marked a point
on the cement. We then measured the length from this point to the first marked pole.
We then used the compass to find the angle the chosen point made with the two
marked poles. We then found the angle the first marked pole made with the selected
point and the other marked pole. This resulted in the creation of an Angle – Side –
Angle triangle. We then used the Law of Sines to calculate the distance between the
two marked poles as shown in Figure 4.
The fourth problem was to find the circumference of the Earth. In order to do this
a little research had to be done. The measurements from Figure 3 were used to find the
angle of the sun. The distance from the Tropic of Cancer to Warren was found by using
an online source. The circumference of the Earth was calculated to be 25,880.68 miles,
while the true circumference of the Earth is 24,901 miles. This answer was off by about
a 1,000 miles. We could have gotten a more accurate calculation if we used better tools
to find the angle of the sun and the distance to the Tropic of Cancer.
6
6
∅ = 𝑆𝑖𝑛−1 (16)
sin(∅) = 16
𝐴𝑛𝑔𝑙𝑒 𝑜𝑓 𝑆𝑢𝑛
360°
=
∅ = 20.5°
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑇𝑟𝑜𝑝𝑖𝑐 𝑜𝑓 𝐶𝑎𝑛𝑐𝑒𝑟
𝐸𝑎𝑟𝑡ℎ′ 𝑠 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
Earth′ s Circumference =
1473.76
20.5°
360°
Earth′ s Circumference = 25,880.68
Figure 6. Calculating Earth’s Circumference
𝐴𝑛𝑔𝑙𝑒 𝑜𝑓 𝑆𝑢𝑛 = 20.5°
Ahmed – Davey – Morley
9/25/14
11A
The figure above shows how the Earth’s circumference was calculated. The
triangle used to find the Angle of the Sun is from Figure 3, and the proportions and
distance to the Tropic of Cancer were found online.
In conclusion, this activity has taught us many lessons. One of the lessons
learned was how useful mathematics can be in real life situations. None of our answers
were exact, but this could have easily been fixed by using better equipment. Using a
more precise tool to measure the angles would have increased the accuracy of our
calculations. Using a more precise measuring tool for the lengths would have also
increased the accuracy of our answers.
The main problem we faced with this activity was that our measurements were
not exact. This resulted in calculations that varied slightly from the true answers. For
example, the shadow of the flagpole could not be fully seen because it ended on the
grass. So we decide to use the length of the shadow that could be seen. This was also
reflected in measuring the angles. The angles were measured using a homemade
compass. This compass used a straw and required one of us to look into it to find the
angle measurement because of this the angles were not exact.
One of the lessons we learned in this activity is that in real life we will not be
given the “numbers”. This activity forced us to measure the lengths and actually perform
calculations from real data that we gathered. These numbers did not come out of a text
book and were not perfect. When working in real life situations the numbers will never
be perfect, but precautions should be taken to assure that the most accurate data is
gathered. We were surprised on how useful mathematics can be in real life. This activity
demonstrates some of the real world applications of mathematics.
Ahmed – Davey – Morley
9/25/14
11A
Works Cited
"Distance between Warren, Michigan, USA and the Tropic of Cancer." Distance
between Warren, Michigan, USA and the Tropic of Cancer. Date and Time
Info, 15 July 2014. Web. 21 Sept. 2014.
Rubin, Julian. "Eratosthenes: The Measurement of the Earth's Circumference."
Eratosthenes: The Measurement of the Earth's Circumference. June 2013.
Web. 20 Sept. 2014.