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lea

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To find the height of a tree, Lina marked the tree at eye level, 5.5 feet above the ground. She measured 25 feet from the base of the tree and then held a 6-inch ruler vertically in front of her eye until the ruler just obscured the tree above the mark. Using a string tied through a whole in one end of the ruler, Lina found that the distance from her eye to the ruler was 4.5 inches. What was the EXACT height of the tree?
 
lea said:
To find the height of a tree, Lina marked the tree at eye level, 5.5 feet above the ground. She measured 25 feet from the base of the tree and then held a 6-inch ruler vertically in front of her eye until the ruler just obscured the tree above the mark. Using a string tied through a whole in one end of the ruler, Lina found that the distance from her eye to the ruler was 4.5 inches. What was the EXACT height of the tree?
Click to expand...

The first example on this page shows exactly how to set this up. :wink:
 
Hello, lea!

Did you make a sketch? .Do you have a game plan?

To find the height of a tree, Lina marked the tree at eye level, 5.5 feet above the ground.
She measured 25 feet from the base of the tree and then held a 6-inch ruler vertically
. . in front of her eye until the ruler just obscured the tree above the mark.
Lina found that the distance from her eye to the bottom of the ruler was 4.5 inches.
What was the EXACT height of the tree?
Click to expand...

I converted all measurements to inches.

Code: 
                              oC -
                          *   |  :
                      *       |  :
              E   *           |  :
              o               |  :
          *   |6              |  : h
     Ao - - - o - - - - - - - oG :
      |  4.5  F               |  :
    66|                     66|  :
      |                       |  :
     Bo-----------------------oD -
      : - - - -  300  - - - - :
The height of Lina's eye is: \(\displaystyle AB = 66\)
The height of the tree is: \(\displaystyle CD = h\)
Lina marked the tree at \(\displaystyle G\!:\,GD = 66\)
Her distance from the tree is: \(\displaystyle BD = 300\)
The ruler is: \(\displaystyle EF = 6\)
Its distance from Lina is: \(\displaystyle AF = 4.5\)

Use similar right triangles to find \(\displaystyle CG.\)
Then find \(\displaystyle h \:=\:CG + 66.\)
 
This is what my work looks like

well what i did was make 2 triangles one bigger than the other. I made the smaller base 4.5 and the larger 25. The smaller height i made 6 and then i cross multiplied got 4.5x=150 divided by 4.5 on each side and got 33.3333. Then i did 33 1/3 + 5 1/2 = 38 5/6 or 38.83. So I am thinking the tree is 38.83. Am I right or not?
 
It is not really necessary to "convert all measurements to inches" because you can set everything in terms of ratios. You have a 6" ruler at a distance of 4.5 inches- that's a right triangle with opposite side 6" and near side 4.5". The ratio is 6/4.5. Looking at the same parts of the "large" triangle, you have the unknown, x, the height of the tree above eye level measured in feet, as opposite side and the distance to the tree, 25 feet, as near side. Since the angles are the same, these are similar triangles and so \(\displaystyle \frac{6}{4.5}= \frac{x}{25}\). Both ratios are "dimensionless" now. Multiplying both sides by 25 feet, we get x= (6/4.5)(25)= 33 and 1/3 feet. Now, add the 5.5 feet to eye level and get your result of 38 and 5/6 feet for the height of the tree. Well done!
 
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