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Do Bees Build it Best?

Is honeycomb the most efficient shape for honey storage?
by

John Tennison

on 9 October 2012

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Transcript of Do Bees Build it Best?

Do Bees Build it Best? Is honeycomb an efficient shape to use for honey storage? We began by studying triangles: If a rectangle has area
equal to base times height... A=b•h b h ...and a triangle is half as big... ...then the area of a triangle must be half the base times the height! A=½ b•h (rectangle) (triangle) Hypotenuse (h) Then came trig!
Some students think this is the hardest branch of mathematics, but when you break it down it isn't so bad: Start by figuring out which angle you care about.
It's never the right angle, but it could be the top or bottom.
In this case lets assume we know the top angle, ø ø The hypotenuse is always the longest side, but once we have our angle we can label the others! opposite (o) adjacent (a) this is the opposite side because
its farthest away from the angle This is the adjacent side because
its the side closer to the angle Once we have our sides labeled we can set up the trig ratios. Remember:
sin Ø = o/h cos Ø = a/h tan Ø = o/a

The way I remember them is sohcahtoa lets do a quick example:
lets say the angle in our triangle is 36º and the hypotenuse is 6 feet long. Which trig ratio would we use to find the adjacent side? its cosine! So we would set up our equation like this:

cos (36º) = a/6 first we plug cos (36º) into a calculator We also found the area of some other shapes: Parallelogram A=b•h (just like a rectangle!) b h Trapezoid A= ½ (b1 + b2) • h h b1 b2 Now for the Pythagorean Theorem! Pythagoras was one of the first mathematicians we have good record of. He lived in Greece around 550 B.C.
Besides being a mathematician he was a philosopher, and even created his own religious sect! we studied a proof in class, but there are a million more! Even one of our presidents, James Garfield, came up with his own. Here is a different proof a²+b²=c² c b a a b lets look at this right triangle.
we'll label the legs a and b and the hypotenuse c c now lets make 4 of them and
put them in a square like this: a b c a b c a b c a b c the sides of the big square are a + b,
so its area must be (a+b)².

the area of the big square in the middle
has sides of c, so its area must be c² now lets make one more square
with those 4 triangles a different way: b a a b a b I drew in these
lines of length a a a and these lines
of length b b b Now, this big square still has sides of a+b, right?
So its area must be the same as the last one: (a+b)²

Both squares have 4 of those same triangles in them.
The last one also had a huge c² in the middle.
What does this one have? Two smaller squares!
the top one has sides a, so it must be a²
the bottom one has sides b, so it must be b²

Since both big squares have the same area, a²+b²=c²! from 2 dimensions to 3! We used these ideas to determine the area
of different polygons: by dividing the shapes up into triangles, we
were able to piece together the area by making right triangles,
we could figure out the
angles and use our handy-
dandy trig to find the triangle's height! A pentagon has 540º (180º more
than a quadrilateral) and 5 corners.
Therefore each corner must be 540º/5 or 108º! Since we have
divided our corner in half,
this angle must be 54º 54º In class we dealt with a perimeter of 300 ft.
To get the side of our right triangle we first divide by 5
to get the side of a pentagon, then we divide that by 2 for our right triangle. That gives us 30! 30 ft 30 ft Now its a simple matter of trig:
we have to use tangent to find the height we want, so we get tan (54º) = h/30. This gives us h= 41.3 ft That means the area of the triangle must be ½•30•41.3

Since there are 10 of these right triangles in the whole shape, we can multiply 619.4•10 to get 6194 ft². Next we tackled Volume and Surface Area of 3D objects.
Boxes are easy: Volume = length • width • height
Surface Area = top + bottom + front + back + left + right
top + bottom = 2•(width•length)
front + back = 2•(width•height)
left + right = 2•(height•length) Remember!
Volume is cubic units (units³)
Area and Surface Area are squared units (units²)
and Lines are just plain units Because of this, when we double linear dimensions
Area gets multiplied four (2²)
and Volume gets multiplied by eight (2³) What happens when we triple lengths?
Area gets multiplied nine (3²)
and Volume gets multiplied by 27! (3³) We tackled other right prisms too! For volume, find the area
of the base and multiply by height For surface area of the sides,
find the perimeter of the base
and multiply by the height.
This brings us to the end of the unit!
Boy have we covered a lot! But wait! What about the bees?
Are hexagonal right prisms really
the best way to store honey?? Well, another thing we talked
with increasing sides, can you answer the unit question? Doing this a few times we learned
that if a polygon has the same perimeter,
the more sides it has the bigger the area! 36º 6ft adjacent .81 = a/6 then we multiply by that denominator a = 4.85 feet!
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