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Thread: Height of Ponderosa Pine

  1. #1
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    Height of Ponderosa Pine

    We've got a very large ponderosa pine on our property. It's got a touch more than a 3 foot diameter at the base. We wanted to figure out how tall it is. I took a photo of Mrs. Cougar down by the base of the tree. She's dwarfed by the size of the thing! We got an estimate using her height as a measure of 5 feet 8 inches.

    tre937.jpg

    Then I thought of "measuring" the height using my golf laser range finder, which is very accurate and currently set to register in yards (100 yards to the hole, that's an A wedge for me. )

    Anyway, I measured 60 yards in a horizontal line to the base of the tree's trunk, then looked up to measure 70 yards to the top of the tree. The 70 yards is obviously the hypotenuse of a right triangle, with 60 yards as the base, and the missing leg is the height.

    So the math operation we want is (702-602)^1/2

    ...which Wolfram Alpha says yields about 36.05 yards or 108.15 feet.

    What I found kind of odd, however, was Wolfram's additional point that the expression (702-602)^1/2 has an exact solution of 10*(13)1/2

    Ten times the square root of 13??!!! (I can hear John Oliver saying it now.) In any case, it seemed quite odd to me that those parameters (70 and 60) would reduce within this Pythagorean expression to ten times the square root of 13!
    Everyone is entitled to his own opinion, but not his own facts.

  2. #2
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    Cougar. Take a meter stick out in the yard when the sun is high. If the shadow the upright meter stick makes is say...75 cm near noon, then the shadow the tree makes is 3/4 of it's real height too.....similar triangles. Measure your tree shadow with the meter stick or a tape, multiply by 4/3, and there you go.....

    pete

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    Quote Originally Posted by Cougar View Post
    Ten times the square root of 13??!!! (I can hear John Oliver saying it now.) In any case, it seemed quite odd to me that those parameters (70 and 60) would reduce within this Pythagorean expression to ten times the square root of 13!
    Well, squares of multiples of 10 are always going to come out to be multiples of 100, as will their sums and differences: 702-602=1300. And the square root of a multiple of 100 will always turn out to be 10 times [some square root]; in this case, the square root of (72-62). So it's those nice round numbers going in that produce the neat exact solution.

    Grant Hutchison
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    Trees have evolved for maximum experienced wind speeds and this may involve tropisms that detect wind loads (survived) and add wood. Or the tree just grows a bigger trunk at the same time as increasing its wind load area. Either way there is a relation between trunk diameter and height. The former seems to apply when we see trees in a group are thinner and taller than free standing ones.
    The second moment of area of the trunk increases roughly with d^4 . Something else to measure and record?
    sicut vis videre esto
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  5. #5
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    I measured the large pondo in my front yard with a golf rangefinder as well. Came up with ~135 feet (it's the tallest tree in our neighborhood). Later that year when we had tree people out to do maintenance on the thing, one of them climbed to the top and dropped a line to measure the height: 142 feet. So my rangefinder estimate was pretty close.

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    Quote Originally Posted by grant hutchison View Post
    Well, squares of multiples of 10 are always going to come out to be multiples of 100, as will their sums and differences: 702-602=1300. And the square root of a multiple of 100 will always turn out to be 10 times [some square root]; in this case, the square root of (72-62). So it's those nice round numbers going in that produce the neat exact solution.
    Well analyzed! Man, I guess I'm really losing my mathematical intuition....
    Everyone is entitled to his own opinion, but not his own facts.

  7. #7
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    It's really just a matter of extracting common factors: 702-602 = 102(72-62)
    Then it's easy to see where the 10 and the sqrt(13) come from.

    Grant Hutchison
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    Another way to measure the height of this tree without an inclinometer is to stand some distance from it while holding a ruler or stick or stick of known length at arms length so that it just coincides with the height of the tree. Measure the distance to the base of the tree. Have someone measure the distance between your eye and the bottom of the ruler. These measurements set up a pair of similar triangles such that Height of the tree = (length of stick)*(distance to tree's base)/(distance between eye and base of stick). This way you don't have to ensure that you've set up a perfect right triangle.

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    Quote Originally Posted by trinitree88 View Post
    Cougar. Take a meter stick out in the yard when the sun is high. If the shadow the upright meter stick makes is say...75 cm near noon, then the shadow the tree makes is 3/4 of it's real height too.....similar triangles. Measure your tree shadow with the meter stick or a tape, multiply by 4/3, and there you go.....

    pete
    Works if the ground is level. If not, may be problematic.
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  10. #10
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    Quote Originally Posted by grant hutchison View Post
    It's really just a matter of extracting common factors: 702-602 = 102(72-62)
    Then it's easy to see where the 10 and the sqrt(13) come from.
    I was struck by the fact that 72-62= 13 = 7+6.
    It turns out to be a general rule for the difference between the squares of numbers that differ by 1.

    (x+1)2-x2 = (x2+2x+12)-x2 = (x+1)+x

    I did not know that.

    Grant Hutchison
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  11. #11
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    Quote Originally Posted by grant hutchison View Post
    I was struck by the fact that 72-62= 13 = 7+6.
    It turns out to be a general rule for the difference between the squares of numbers that differ by 1.

    (x+1)2-x2 = (x2+2x+12)-x2 = (x+1)+x
    genie.gif
    Everyone is entitled to his own opinion, but not his own facts.

  12. #12
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    Trebuchet.yep.needs to be level for similar triangles...and the measure should be from the center of the meter stick bottom and the center of the tree bottom to the tip of their shadows on the ground. pete

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    I like big trees, I cannot lie!
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