Purpose: real life application of trignometry

2. Purpose: real life application of trignometry Physicists and engineers utilize vectors and trig for solving a variety of problems related to relative velocity, Motion in 2 Dimensions and Forces in 2 dimensions. However, even foresters use trignometry.

3. Introduction:  • Scientists studying a forest ecosystem over a long period of time may record measurements of trees for a number of variables, including each tree's diameter at breast height, height of the lowest living branch, canopy cover, etc. One aspect of a tree's growth that can be hard to measure is tree height.

4. Forest researchers sometimes use a piece of equipment consisting of telescoping components, which are extended until the tip reaches the same height as the tree top (this requires a second researcher standing at a distance from the tree to determine when the tip is at the correct height). This method can be cumbersome, as the equipment is bulky and the measurements require two people.

5. Importance of measuring tree height • measurement of tree growth in a forest over time provides important information about the dynamics of that ecosystem. • Growth rates can reflect, among other things, differing availability of water, carbohydrates, or nutrients in different sites or from season to season or year to year.

6. Question:  Is there an efficient way to measure tree height, without heavy equipment and multiple people?

8. The angle of importance • While holding the inclinometer, you will be standing at angle C. • Measure the distance you are standing away from the base of the object you're finding the height of, which is segment BC.

12. If you don’t want to take into account your height from eye level to the ground – you can always do this…

13. Challenge: • A surveyor wanted to measure the height of a mountain. • He traveled in the straight road leading to the mountain and measured the angle of elevation of the peak from a point A which is on level ground with the mountain base as 32º. • He drove 1 Km further down the road and again measured the angle of elevation of the peak as 40º.  • Find the height of the mountain nearest to a meter.

14. Solution • As we see in the pictures, two variables h and x are introduced, the height of the mountain and the distance BC. Using trig formulas we can eliminate x and solve for h.  In right  triangle BCD,

16.   tan 40 = hx   ⇒       x = htan40   in right triangle ACD,    tan 32 = hx+1              = hhtan40+1                   Substitution             = htan40h+tan40                      Complex fraction simplified.  (h + tan 40) tan 32  = htan 40                              Cross multiplicationhtan 32 + tan 40.tan 32 = htan40htan 40 - htan 32 = tan 40.tan 32    h(tan 40 - tan 32) = tan 40.tan 32   h = tan40tan32tan40−tan32       ≈ 2.4475 Km   Hence the estimated height of the mountain = 2,4475 meters.

18. Pre-lab Questions: 1. What are you looking for? 2. How will you solve for it?

19. Use Metersdon’t copy this – use your own data and object. 2 Meters 2 Meters Meters