Hexagon Sides, Length of a Beam
Date: 1 Feb 1995 20:27:58 -0500 From: Cheryl A. Backman Subject: math q Hi again. I am totally stumped on these two word problems. They are driving me crazy! Here goes: 1. Determine the length of each side of a regular hexagon if opposite sides are 18 in. apart. 2. A wooden board is placed so that it leans against a loading dock to provide a ramp. The board is supported by a metal beam perpendicular to the ramp and placed on a 1 ft. tall support. The base of the support is 7 feet from the point where the ramp meets the ground. The slope of the ramp is 2/5. Find the length of the beam to the nearest hundredth of a foot. Please help! Could you possibly respond by 11p.m. this evening or 6:30 tomorrow morning? Thanks. Michele
Date: 3 Feb 1995 18:22:35 GMT From: Dr. Math Subject: Re: math q Dear Michele, Sorry we weren't able to respond as quickly as you requested -- we are pretty busy right now, so it is taking us a few days to answer questions. Let's look at your questions now, though -- better late than never, eh? 1. A regular hexagon has 6 equal sides that are at 120 degree angles with one another. To do your problem, I think it helps most to draw a picture. Since I am not very good at drawing on the computer, you'll have to draw your own picture on a piece of paper. Okay, draw a regular hexagon. Now, find a vertex in the hexagon (it doesn't matter which one, since the hexagon is regular) and call it point B. Find the points corresponding to the vertices that are next to this vertex and call them A and C. Then, draw a line connecting A and C-- you should now have a triangle with a 120 degree angle at point B. The length of the side of the triangle opposite point B is going to be 18 inches. Can you see why from your picture? The line from A to C is the line connected opposite sides. So, now, draw a line from B to the middle of line AC so you have two identical right triangles. Call this new vertex D. Now look at triangle ABD. The angle at vertex B is half of what it was before we drew the line to D, so it is now 60 degrees, while the angle at D is 90 degrees. Since the interior angles must add up to 180 degrees, angle A must be 30 degrees. This makes things pretty nice because 30-60-90 triangles have some nice properties. In particular, in 30-60-90 triangles, the side opposite the 90 degree angle is 2/sqrt 3 times as long as the side opposite the 60 degree angle. That means that in our picture, side AB is 2/sqrt 3 times as long as side AD. In our picture, side AD is half as long as side AC which was 18", so side AD is 9". This means side AB has length (9")2/sqrt 3. Side AB is one of the sides of the regular hexagon, so the regular hexagon that has sides 18" apart has sides of length 18"/sqrt 3. 2. I'm not completely clear on the wording of this problem, so maybe you could write back and tell me if I am understanding it correctly. I drew a picture, and here is roughly what it looks like. Is it right? / / / / /\ /90\ x / \ / \ | / -7ft- 1 ft. -------------|----------------------------- I know the picture is not in correct proportions. The 90 indicates a 90 degree angle, the -7ft- means the horizontal distance on the ground there is 7ft, and the | 1 ft means 1 ft in between the ground and the beam | You are trying to find the distance x, given that the line is rising with a slope of 2/5, right? Write me back to tell me if I have the problem right and I will help you work through it, okay? Hope this helps. --Sydney, "dr. math"
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