Brain Teasers
Old Oak Tree
Homer had always contemplated the size of the old oak tree in the field behind his house. Standing 5 feet away from the 8 foot high fence in the back of the yard, he could just barely see the top leaves of the tree over the top of the fence. He knew that the tree was 200 feet from the fence. Homer was 6 feet tall, making his line of vision, 67 inches from the ground.
Estimate how tall the old oak tree is.
Estimate how tall the old oak tree is.
Hint
TAN A = opposite / adjacent, remember trigonometryAnswer
tree height = TAN of angle of vision * distance to the tree = 104.7 feet tallFirst you need to figure out the angle of vision. The angle of the line of vision over the fence, is the same angle of vision of the eye to the tree top. Height of eyes (67 inches), height of fence (8 ft = 96 inches), distance to fence (5 ft = 60 in)
Angle of vision = inverse TAN ( (96 - 67) / 60) = 25.80 degrees
Distance to tree = 5 ft + 200 ft = 2460 inches
Tree height from eye level = TAN (25.80 deg) * 2460 inches = 1189 inches
Tree height estimate = 1189 inches + 67 inches = 104.7 feet
This is an old lumberjack method of estimating a tree height. With pure math the height of 104.7 feet is correct. With the measurement capabilities available to any lumberjack (or any average Homer) precise measurements are not possible. Thus the need to clarify these calculations as estimates.
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Comments
I did not get how to solve this one, i am not a real math wiz.
I am humbled by my limited math knowledge.
Intelligent teaser
Intelligent teaser
next time tell us about SohCahToa, that'll help alot, i don't know anything about trig, this was geometry
Very creative teaser. Allthough math and I seem to keep a wide distance between each other, I sitll enjoyed reading your answer and following some of it!
You don't need trig to do this problem - its a simple proportion.
Considering the heights above eye-level the height of the tree (h) compared to the distance from the observer (205) = height of the fence (2 5/12) to distance from observer (5). Thus h/205 = 2 5/12 / 5) or 5h = 205 * 2 5/12. This gives a value of h = 99 1/12 feet. But this is the height above eye- level so adding back the eye height we have Tree = 99 1/12 + 5 7/12 = 104 2/3 feet. A junior high proportion problem, but that's OK 'cause I like Mathematics!
Considering the heights above eye-level the height of the tree (h) compared to the distance from the observer (205) = height of the fence (2 5/12) to distance from observer (5). Thus h/205 = 2 5/12 / 5) or 5h = 205 * 2 5/12. This gives a value of h = 99 1/12 feet. But this is the height above eye- level so adding back the eye height we have Tree = 99 1/12 + 5 7/12 = 104 2/3 feet. A junior high proportion problem, but that's OK 'cause I like Mathematics!
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Huh?
Aug 28, 2005
Draw a picture of the fence and homer (up to his eyes) and label it with the heights of each in inches. Then draw a triangle at the top of the two lines, which has a base of 60 inches (5 feet away from the fence) and a height of 29 inches (Fence - Eyes). You don't even need to find out the hypotenuse, just extend this same triangle out an extra 2400 inches towards the tree (200 feet). The new triangle from Homer's eyes to the top of the tree has a base of 2460 inches (Homer to fence, fence to tree), and a height of H. If a base of 60 makes a height of 29, then a base of 2460, which is 41 times that amount, makes a height also 41 times the original. 41*29 = 1189 inches.. which (if you really have been drawing the picture) you can see is the height from Homer's eyes' to the top of the tree. Add homer's height and you get the answer of 1256 inches, or 104.66 feet
Thanka for that good explaination. It gives a good visual.
When I saw all the trig in the solution I thought, "Wow, I did that problem in my head and I can't do arc tangent in my head!"
Of course this should be done as a similar triangle as pointed out above.
Of course this should be done as a similar triangle as pointed out above.
hah yeah just use similar triangles...it's pretty easy...even though i did mess it up and got 84.7 ft instead of 104.7 ft...but i just messed up the subtraction of 67 inches from 96 inches (8 feet)...i had it as 23 inches (was doing it as 90 - 67 for some odd reason idk why)...and that threw off my whole ratio...i quickly realized my mistake though...
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