Inquiry Question Can I Build a Theodolite to Calculate the Height of a Building?

Inquiry Question Can I build a theodolite to calculate the height of a building?

Name: ______Date: ______

General Instructions

Theodolites are tools that allow us to collect information about land (like angles and heights) and are used extensively for land surveying, the building of structures, and in some unique cases for meteorology (weather) and rocket launching! You will build a simple theodolite to measure a couple angles, then use that data to determine the height of a building or structure.

Materials you’ll need:  Pencil, eraser, scientific calculator, ruler/tape measure  To build the theodolite: o Scissors, glue, protractor (can print one), compass (for drawing circles), cardboard, thread/string, paper, small object, brad/paper fastener

Project submission: Submit the completed pages of this project, showing any work you did. Also include a clear picture of yourself with your completed theodolite and the object you determined the height of.

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Math: Angles

Background Have you ever looked along the side of the road in an area of construction and seen a yellow object on a tripod with someone looking through it? This object is typically called a theodolite and is used by land surveyors to determine very accurate locations of points. Modern theodolites measure angles and distances electronically, but simple theodolites have a telescope view-piece that can rotate horizontally (side to side) and vertically (up and down) and allows you to determine both angles.

Building a Simple Theodolite Your first task is to build a simple theodolite. It only needs to be able to measure the vertical angle of inclination and not the horizontal angle. To build your theodolite, see the following webpage for instructions: https://www.instructables.com/Simple-Theodolite/ This video shows a similar method to making a simple theodolite, though you will need a protractor and slightly varied materials: https://www.youtube.com/watch?v=FVqNEBWH4B0 Printable protractor (would need to mount on cardboard): https://www.cool2bkids.com/wp-content/uploads/2019/08/Printable-Protractor.pdf

Both the webpage and video also contain instructions on how to use the theodolite. Refer to these if you need a reminder later in the project.

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Taking Measurements You will find the height of an object that is on level ground and has an inaccessible base. This strategy can be used to determine the height of a distant object without having to be near it (that said if you pick an object you can touch like your house that is okay). Below is a diagram that includes the variables you will either measure or be asked to calculate.

Your goal is to determine the value of FG to find the total height, EG = EF + FG.

1) First, identify all parallel lines and record them in the space below.

Head outside and determine the values of the following measurements and record them below: 2) The height of the theodolite when taking measurements from point A: o AC = ______meters 3) The angle of inclination when looking at point G from point C: o = ______degrees

4) Move휃 closer to the object. This distance should be greater than or equal to 5m. o d = AB = ______meters 푑 = 퐴퐵 5) The height of the theodolite when taking measurements from point B: o BD = ______meters

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Math: Angles

6) Does BD need to be the same height as AC? Why or why not?

7) The angle of inclination when looking at point G from point D: o = ______degrees

∅

Analysis Questions

8) Determine the values of the green angles, and , given: and (measured above), and the parallel lines (identified above in #1). Show your work below and include reasons for each of your calculations, like as in a proof.훼, 휔, 훾 휃 ∅

9) What type of triangles are both GCF and GDF? ______10) What types of angles are the following:

o GDF ______o CDG ______

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Math: Angles

Calculation of Objects Height To determine the height FG, we need to use trigonometry and some algebra. The steps are shown below: Determining height of GF

…(step 1)… using trigonometry 퐺퐹 푡푎푛휃 = 퐶퐹 …(2)… cross multiplying 퐺퐹 퐶퐹 = 푡푎푛휃

…(3)… using trigonometry 퐺퐹 푡푎푛휙 = 퐷퐹 …(4)… cross multiplying 퐺퐹 퐷퐹 = 푡푎푛휙 We don’t know the length of CF or DF, but we do know that CF – DF = d

…(5)…

퐶퐹 − 퐷퐹 = 푑 …(6)… by substituting the equalities in (2) and (4) 퐺퐹 퐺퐹 푡푎푛휃 푡푎푛휙 − = 푑 …(7)… factoring out GF 1 1 푡푎푛휃 푡푎푛휙 퐺퐹 ( − ) = 푑 …(8)… isolating GF by dividing 푑 1 1 1 1 퐺퐹 = 푡푎푛휃−푡푎푛휙 푡푎푛휃 − 푡푎푛휙 11) Using the equation found in (8), determine the height GF. Show your work below.

12) What is the total height of your object EG? Show your work.

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