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Felling a Tree - February 5-9, 1996 Next Problem || Previous Problem || Contents || All Problems || Search POWs As part of the never-ending ordeal of homeownership, there are a bunch of things I need to do in the yard. Among them, there's a tree in the back right corner of the yard that needs to come down - it's not an attractive tree, and it's shading the garden. I need to drop the tree between the veggie garden and the willow tree, but if it's any taller than 59 feet I'll have a problem, because there's a lilac bush that I don't want to smash. To figure out how tall the tree is, I went out today and measured the tree's shadow and my shadow when I stood next to the tree. The tree's shadow was 76'8", and mine was 94 inches. Given that I'm 5'10", will the tree land on the lilac or not? Now, consider the fact that in my work boots I'm probably closer to 6 feet tall. How does the situation look now? - Annie Fetter Solutions Annie says: We had a whole lot of correct solutions this week and I will admit to being a little tough on those who got only the second part right. While it is true that, to know whether the lilac gets smashed, you really only need to do the second part where I am "taller," I did not give credit to those who simply did the second part and didn't say why they didn't have to do the first part. The question poses two problems, so you should solve both. (My favorite reason why one team didn't do the first part was from Sara Holtzman and Jenny Shaefer: "We didn't figure it out without your shoes on because most people wouldn't go outside with no shoes in the dead of winter.") While most people used similar triangles, a few people used trig to solve this, figuring out the angle of elevation and going from there. There are some good explanations along with those solutions. As to whether it is actually safe to cut the tree down, that is a matter of opinion. Once you find a numerical answer to a problem, you have to decide what it really means. If I am indeed 6 feet tall in my workboots, that only leaves a few inches to spare, and several people said that they didn't think that was safe. For some people, 3.5" is enough leeway; for others it isn't. I'm cutting the tree down - I padded the distance to the lilac a bit anyway, so it's a little bit more than 59 feet, and I'm pretty handy with a chainsaw. :-) We welcome four new schools this week, including one from South Australia and our first school from New Hampshire. Following are highlights, and the names of all the people who submitted correct solutions and most of the solutions are also available. Give the submissions a good read - while most people used the same method, the solutions vary greatly in length and clarity. See which ones you think are effective and really explain how to approach the problem. Christine Francescani and Julie Conant Grade 10 Martin County High School, Stuart, Florida To begin, we drew the diagram and labeled points on the two triangles A-E. We said that triangle BAC is similar to triangle EAD because corresponding angles are congruent, and the corresponding sides are in proportion. We then said that side CA is to side CB as side DA is to side DE. Therefore, 94" is to 70' as 920" is to X. So, X equals 685.11", which is 57.11'. If you are 6' tall, then the tree is 58.7' tall. In either case, the tree will not hit the lilac bush. Dorothy Moorefield Grade: 11 School: Walter Williams High To solve this problem, use the fact that the angles formed from revolving from the ground to the top of Annie and the tree (x) are congruent. Two similar right triangles are formed. tree Annie /l /l (y^2+920^2)^0.5" / l 13736^0.5" / l 70" / l y" /x l /x l ------- --------- 94" 920" The lengths of the bases of the triangles formed are the lengths of the shadows converted to inches. The height of the tree is unknown so it is called y. Annie's height is 70 inches. The lengths of the hypotenuses are found using the Pythagorean theorem [leg1^2 + leg2^2 = hypoteneuse^2] Set these triangles in circles whose radii are the hypotenuses to find the length of y. Since x = x the cos x = cos x. The cos x = (the length of the adjacent side)/(the length of the hypotenuse) cos x = 920/((y^2+920^2)^0.5) cos x = 94/(13736^0.5) 920/((y^2+920^2)^0.5 = 94/(13736^0.5) 11626150400 = 8836y^2 + 7478790400 4147360000 = 8836y^2 4147360000/8836 = y^2 y is approximately = 685.106" approximately = 57' So the tree could come down within 2 feet of hitting the lilac bush. If Annie is closer to 6' plug in 72" for her height and repeat the process. DO NOT FORGET TO CHANGE THE LENGTH OF THE HYPOTENUSE. cos x = 920/((y^2+920^2)^0.5) cos x = 94/(14020^0.5) 920/((y^2+920^2)^0.5) = 94/(14020^0.5) 11866528000 = 8836y^2 + 7478790400 4387737600 = 8836y^2 4387737600/8836 = y^2 y is approximately = 704.681" approximately = 58.724' If Annie is closer to 6', the tree could come down but it would be cutting it awful close. An easier way to solve this problem would have been to set up proportions because the triangles are similar. However, the cos x method was more fun to explain. Sara Holtzman and Jenny Schaefer Grade 9 Martin County High School, Stuart, Florida First we must convert feet into inches. Six feet converts to 72 inches. 76'8" converts to 920 inches. Then we set up a proportion 920":94" = x:72" reducing it to 460":47" = x:72". Multiplying it out we came to the equation 33,120" = 47"x. Dividing the equation we came to the conclusion that x = 704.68085". Converting this back to feet we got 58.723404 feet, which rounds off to 59 feet. This tells us that the tree would just miss the lilac bush. We didn't figure it out without your shoes on because most people wouldn't go outside with no shoes in the dead of winter. /tree / |/ / | / | / | you\ / | /| | / |6" | / | | ------------------- |------| 94" |------------------| 76'8" Therese Quinn Year 10 Loreto College, Marryatville, South Australia !\ ! \ ! \ ! \ ! \ ! \ ! \ ! \ Height ! \ of ! \ Person ! \ 70" ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ !____________________\ Length of shadow (a) To figure out the height of the tree, I first worked out the sun's angle of elevation by looking at the height of the person and the length of her shadow, and using the trigonometry function 'tangent'. tan a = opp --- adj = 70 -- 94 tan 36.67 = 70 -- 94 The sun's angle of elevation is 36.67 degrees. Because the shadow of the tree was taken at the same time as the person's, the sun's angle of elevation would have been the same. I used the tangent function once again to figure out the height of the tree. (76'8" = 920") tan 36.67 = height of tree -------------- 920 height of tree = tan 36.67 X 920 = 685" = 57'1" Therefore the tree wouldn't squash the lilac bush when it fell. IN WORKBOOTS, THE PERSON IS LIKELY TO HAVE BEEN CLOSER TO 6'. If this were the case, the numbers would have been different. tan a = opp --- adj = 72 -- 94 tan 37.45 = 72 -- 94 Therefore, the sun's angle of elevation is 37.45 degrees tan 37.45 = height of tree -------------- 920 height of tree = tan 37.45 X 920 = 705" = 58'9" In this case, the tree would only miss the lilac bush by 3 inches. Personally, I wouldn't risk it, but then, if the tree's really that ugly, and if it's shading the garden... Karen Wing Grade 10 Mt. Saint Joseph Academy, Flourtown, PA Because you are 5'10" and your shadow is 94", by taking the inverse tangent of 70" / 94", you can find that the angle of the shadow is 36.7 degrees. Since you were standing next to the tree the triangle between you, the end of your shadow, and the top of your head is proportional to the triangle of the tree, the end of its shadow and the top of the tree. Therefore the angle at the end of the tree's shadow is also 36.7 degrees. Then take the tangent of 36.7 degrees and multiply it by the 76'8" or 920", the length of the shadow. The answer will be the length of the tree, which is 57'2". Therefore the tree will not hit the lilac bush but will actually be 1'10" short of hitting it. Since you are 6' in your boots or 72" and your shadow is 94", by taking the inverse tangent of 72"/94" we find that the angle is 37.5 degrees. By using the same two proportional triangles you can say that the angle at the end of the tree's shadow is now 37.5 degrees. By taking the tangent of 37.5 degrees and multiplying it by 920" again, we find that the length of the tree is now 57'9". This length of the tree will also be short of hitting the lilac bush by 1'3". Katie Walder Mount St. Joseph Academy, Flourtown, PA After looking at this problem, I decided to set up a proportion. I set your shadow over the tree's shadow, which was 94/920, and I set that equal to your height over the tree's height, which was 70/h. (I used your height in your workboots first because if it fell without hitting the lilac bush then, it must also fall without hitting it when you are at your normal height.) 94/920 = 70/h 920 * 70 = 94h 64400 = 94h h is approximately 685 in. or 57.09 ft. Since it falls right when you have your workboots on, it must also fall right when you don't. Anna Margush 4th grade, age 9, home school Using similar triangles the tree is 685.1 in or 57.1 ft. The lilac is safe! If you are really 6 ft. (72 in.), then the tree would be: 72 ? ---- = ---- 94 920 ? = 704.7 in = 58.7 ft It still would not flatten the lilac, but it would be very close. Laura Ejups Grade 10 Martin County High School, Stuart, Florida Using the proportion 6:7.83=76.67, I figured out that x = 58.75. This told me that the tree would miss the lilac. You can cut the tree when you are 5'10'' and when you are 6'0''. In both cases it won't land on your lilac bush. Work: 5.83 x 6.0 x ----- = ----- ------ = ----- 7.83 76.67 7.83 76.67 x = 57.09 x = 58.75 at 5'10'' /| x= 57.09 / | at 6'0'' / | x= 58.75 / | / | / | / | / | < TREE / |< YOU | / | | / | 6'0'' OR | / | 5'10'' | ------------------------ |-7'10''| |--------76'8''---------| [Privacy Policy] [Terms of Use] Home || The Math Library || Quick Reference || Search || Help  © 1994-2008 Drexel University. All rights reserved. http://mathforum.org/ The Math Forum is a research and educational enterprise of the Drexel School of Education.