Review

demha · 2013-09-23 08:24:39

My lesson review. Need help checking if answers are correct and getting the right answers

1. If a hexagon has a side of 3 units, what is the area of the hexagon?
3^2 x 6 / 4 x tan(180/6)
54 / 4 x tan(180/6)
54 / 2.3094
Answer: 23.38

2. If a hexagon has an area of 100 units, what is the length of one side?
100= 3√3 / 2 * s² | s = side
s^2 = 100 / 3√3 / 2
s^2 = 100 / 2.598076
s^2 = 38.490021
Answer: s = 6.204032

3. If a hexagon has a radius (center to point of angle) of 6, what is the side of the hexagon?**need help

4. If a hexagon has a radius (center to point of angle) of 6, what is the area of the hexagon?
Finding the area of ONE triangle:
r^2 sqrt3 /4
6^2 sqrt3 /4
36 x sqrt3 /4
36 x .433021
15.588756

a = 6 x 15.588756
Answer: 93.53

5. If a hexagon is resting on a flat side, and has a total height of 18, what is the length of each side of the hexagon?**need help

6. If a hexagon is resting on a flat side, and has a total height of 18, what is the area of the hexagon?**need help

7. Problem solver (worth 4 points): Come up with a way to find the area and volume of a football. Include in your answer a way to acquire any necessary measurements without cutting or otherwise destroying the football. Also include all necessary formulas to implement your idea. (You don't need to find actual numbers, just outline the method in step by step detail--think of all the measurements you'll need to acquire and how you'll get them.) **i have an idea for this but would like to post it after finishing up #1 - #6**

Last edited by demha (2013-09-23 08:31:50)

anonimnystefy · 2013-09-23 08:36:21

demha wrote:

My lesson review. Need help checking if answers are correct and getting the right answers
1. If a hexagon has a side of 3 units, what is the area of the hexagon?
3^2 x 6 / 4 x tan(180/6)
54 / 4 x tan(180/6)
54 / 2.3094
Answer: 23.38
Correct!
2. If a hexagon has an area of 100 units, what is the length of one side?
100= 3√3 / 2 * s² | s = side
s^2 = 100 / 3√3 / 2
s^2 = 100 / 2.598076
s^2 = 38.490021
Answer: s = 6.204032
Correct!
3. If a hexagon has a radius (center to point of angle) of 6, what is the side of the hexagon?**need help
Assuming it is a regular hexagon, the radius is equal to the side of the hexagon.
4. If a hexagon has a radius (center to point of angle) of 6, what is the area of the hexagon?
A = 4(PI)r^2
A = 4(PI)6^2
A = 4(PI)36
A = 144(PI)
Answer: 144(PI) or 452.389
See previous question.
5. If a hexagon is resting on a flat side, and has a total height of 18, what is the length of each side of the hexagon?**need help
Split the hexagon into 6 equilateral triangles. You should notice that the height of the hexagon is twice the height of a triangle.
6. If a hexagon is resting on a flat side, and has a total height of 18, what is the area of the hexagon?**need help
See previous question.
7. Problem solver (worth 4 points): Come up with a way to find the area and volume of a football. Include in your answer a way to acquire any necessary measurements without cutting or otherwise destroying the football. Also include all necessary formulas to implement your idea. (You don't need to find actual numbers, just outline the method in step by step detail--think of all the measurements you'll need to acquire and how you'll get them.) **i have an idea for this but would like to post it after finishing up #1 - #6**
I would have to think about this one, but, as you said, we'll leave it for after the questions above are solved.

Last edited by anonimnystefy (2013-09-23 08:37:24)

bobbym · 2013-09-23 08:36:26

Hi demha;

demha · 2013-09-23 10:27:46

3. If a hexagon has a radius (center to point of angle) of 6, what is the side of the hexagon?
Answer: 6
If the radius is 6, so are the sides.

NOTE: i have edited my first post for #4 but it seems as if you quoted my text BEFORE I edited them. See my new answer:
4. If a hexagon has a radius (center to point of angle) of 6, what is the area of the hexagon?
Finding the area of ONE triangle:
r^2 sqrt3 /4
6^2 sqrt3 /4
36 x sqrt3 /4
36 x .433021
15.588756

a = 6 x 15.588756
Answer: 93.53

anonimnystefy · 2013-09-23 12:34:07

Hi demha

Thise answers are correct!

This means that only Q5 and Q6 are left. Do you want to try it with the hint or do you want me to help a bit more?

demha · 2013-09-23 13:06:19

No I think I got it.

5. If a hexagon is resting on a flat side, and has a total height of 18, what is the length of each side of the hexagon?
A = 9^2 x 6 * tan x (180/6)
A = 280.59 (answer for #6)

Now I have the area, I will solve for S = side of the hexagon:

3 (√3 /2) x S^2 = 280.59
S^2 = 280.59 x 2 / 3 x √3
S^2 = 561.18 / 5.196152
S^2 = 107.99
Final Answer:
S = 10.39

6. If a hexagon is resting on a flat side, and has a total height of 18, what is the area of the hexagon?
A = 9^2 x 6 * tan x (180/6)
Final Answer:
A = 280.59

I'll be doing #7 tomorrow since it will take me a little research and time. Right now I need to get some shut eye. Thank you for helping me out so far!

anonimnystefy · 2013-09-23 13:47:33

Hi demha

Those are the correct naswers!

And no problem! Glad to be able to help.

Just one thing: an easier way to get the side of the hexagon from the height is to set it up like this:
S*sqrt(3)/2=9.

Last edited by anonimnystefy (2013-09-23 13:48:20)

demha · 2013-09-23 22:49:52

7. Problem solver (worth 4 points): Come up with a way to find the area and volume of a football. Include in your answer a way to acquire any necessary measurements without cutting or otherwise destroying the football. Also include all necessary formulas to implement your idea. (You don't need to find actual numbers, just outline the method in step by step detail--think of all the measurements you'll need to acquire and how you'll get them.)
Answer:
I need to find the area and volume of a football without destroying the ball. To do this, I would first start with measuring the ball from top to bottom. I would mark a line exactly on the middle of the ball. Once I have done that, I'll use paper to cast a mold on exactly half of the ball. I will then take it off to have a half football mold which would look like a cone. With this cone, I would find the measurements required to fit into these two equations:

T = (PI)rs + (PI)r^2 - for the cone area
V = (1/3)(PI)(r^2)h - for the cone volume

One I have gotten the answer for each equation, I will simply just x2 both of them to find the final answer of what the volume and area of the football is.

Bob · 2013-09-24 00:25:18

hi demha

What sort of football are we talking about?

A soccer football is roughly a sphere and there are formulas for both the surface area and the volume.

An American football is roughly a prolate spheroid and, again, there are formulas for this. (My third picture below is a prolate spheroid.)

I wonder what is expected for 4 marks ?

There is also a way you can get the volume very accurately with just a large bucket and a measuring jug. Once you have this, you could work back to get the measurements needed for the surface area.

Your cone area method includes the area of the base. When you join two cones back together to get the ball you don't need this value.

And your 'write-up' must include an explanation of what r, s and h are on your paper version.

Bob

demha · 2013-09-24 02:31:56

Yes, I believe they are talking about an American Football, not a Soccer Ball.

For the first equation: T = (PI)rs + (PI)r^2 | solving for area
r = radius
s = slant hight

For the second equation: V = (1/3)(PI)(r^2)h | solving for volume
r = radius
h - height

Would my method be considered a pretty accurate method? Would YOU accept this method?

Bob · 2013-09-24 03:24:02

hi demha

For the first equation: T = (PI)rs + (PI)r^2 | solving for area

The term in red is the area of the base of the cone. When you double your answer you'll end up counting this area (twice) as part of the outside of the ball.

r = radius
s = slant hight

For the second equation: V = (1/3)(PI)(r^2)h | solving for volume
r = radius
h - height

Would my method be considered a pretty accurate method? Would YOU accept this method?

The cone has straight slant sides whereas the ball is curved so there's some inaccuracy there. I would be happy if you discussed why your answer is only approximate as part of your analysis. That way you get credit for realising where the inaccuracies lie. I doubt anyone could get the exact answer from a formula because the ball is pumped up and that makes it virtually impossible to say exactly what shape it is. That's why I said the prolate spheroid is also only a rough shape for the ball.

Another thing you could do is to find an answer that is certain to be too big, and another that is certain to be too small, and then you know the real answer is somewhere between them. At least that makes any inaccuracy clear.

Bob

demha · 2013-09-24 04:16:34

I am starting to get confused as to what methods to use to get the area and volume of the football. Now that I see it, the Prolate Spheroid does resemble the football best. I have never done measurements based on this shape and therefore don't have enough knowledge on how to get the answers (although they might not be accurate, but close). I have found a picture online and came up with another idea. Look at the picture below and see how it is cut up into 4 triangles similarly to what you could do with a perfect circle.

This is my idea on how to get the answers. I would trace the football in a piece of paper and similar to the image, I would draw two lines forming width and height. I would then use measurements finding the slant height, height (pink line as seen on image) and width(green line as seen on image) of the football. Am I going the right way with this so far? Kind of tough for me to figure out what to do. I would appreciate some help understanding what to do and how to do it.

Last edited by demha (2013-09-24 04:18:06)

bobbym · 2013-09-24 04:45:57

Did you see my hint in post #3?

demha · 2013-09-24 05:12:29

Ah I see, that is for the volume, yes?

I see it as:

V f = 4/3(PI) x ab^2

Not sure what the f is, but a = area and b = base, correct?

Bob · 2013-09-24 05:34:15

http://en.wikipedia.org/wiki/Prolate_spheroid

a is half of the distance between the 'ends'. This is called the major axis.

b is half of the length of the minor axis. (how fat it is in the middle)

There is a device called a pair of calipers that would allow you assess these two lengths. You put the points at the end of the legs on the ball and then carefully move the calipers across to a ruler.

The formula for the surface area is much more complicated. You can find it here:

http://en.wikipedia.org/wiki/Spheroid

Fortunately, these is a site that will do the calculation for you:

http://planetcalc.com/149/

A spheroid is a special case of an ellipsoid with b = c.

For your course, I think it would be reasonable to say you would use the on-line calculator.

You have spent quite a bit of time researching ways to carry out the task. I suggest you write up all of them, saying which you think would be easiest to do and which would be most accurate. That way you ought to get top marks.

Bob

demha · 2013-09-24 05:59:37

Oh alright, I see the images on the Wikipedia.

$9550bda48f7f5cfaf523758e0e2fc4db.png$

For the Surface Area.

$925fe567fce60e8a19a5d375ea05080a.png$

For the Volume.

Two questions:
First, what does the letters 'e' and 'c' stand for?

Second, so after finding the major axis (a) and the minor axes (b), I can do the equation?

bobbym · 2013-09-24 06:07:54

Not sure what the f is, but a = area and b = base, correct?

Sorry, mathematicians love to confuse people with subscripts and then superscripts. When my brother tried to teach me Tensor Calculus with all those sub and superscripts, I quit math and ran for public office. That is read V sub f, f stands for football so

is just a label or name. It means volume of the football. a and b are exactly what is in your drawing and mine, they do not stand for area and base. They are axes. But this is all covered in Bob's post.

demha · 2013-09-24 07:36:07

Ok, let me do #7 again. I don't need to find EXACT answers, just find what I think may be the best method to find the area and volume of a football.

I need to get the area and volume of the football. The shape that resembles the ball most would be the Prolate Spheroid. I have two equations that can give me the volume and area of the spheroid. They may not be entirely accurate but they can get me close.

To get the area, I first must do:
e^2 = 1 - a^2/b^2

I need to solve for e. a = major axis and b = minor axis and c = b. To find the two axis, I will need to use a tool called caliper. With the "legs" of this tool, I can get the size of the axis which I will then set beside a ruler to find the measurements. Now once I have finished the equation above, I will then use this final equation which will give me the true answer (or at least the most accurate):

PSA = 2(PI)a^2 (1 + c/ae[sin^-1] e)
PSA means Prolate Spheroid Area

Now that I have my self the area, I will now get the volume using this following equation:

V = (4[PI]/3)a^2 x c OR V = 4.19 x a^2 x c

Now do you think this is ok? If not, what am I missing here? Also, do you think I should send in my previous method with this one too?

Last edited by demha (2013-09-24 07:37:08)

Bob · 2013-09-24 10:26:44

Yes, I would include first the cone method, then say it may lack accuracy because .... (give reason here) .... and go on to the spheroid method.

For this you might as well replace all the 'c' instances with 'b' as it looks like you don't know whether to use c or b. Why might this still not be entirely accurate?

After all that you ought to get full marks (well I would give you full marks so they're very rotten bounders if they don't)

Bob

demha · 2013-09-24 13:18:35

Alright, I sent in the lesson and now I am waiting for the reply which may take a little time. While that is being checked, I did another lesson and got 17 out of 20. Here are the three I got wrong along with the answers I put:

I need to put these points on a graph to answer the questions.

1. Draw a Cartesian Coordinate system for yourself, and then determine whether these sets of points form triangles - I said F

Set 1: (0,0), (1,1), (0,1)
Set 2: (-1, 0), (1,1), (3,2)
Set 3: (0,0), (5,0), (3,5)
A set 1 line segment, set 2 Triangle, set 3 line segment
B set 1 Triangle, set 2 Triangle, set 3 line segment
C set 1 Triangle, set 2 line segment, set 3 Triangle
D set 1 line segment, set 2 line segment, set 3 line segment
E set 1 Triangle, set 2 line segment, set 3 line segment
F set 1 line segment, set 2 Triangle, set 3 Triangle

2. Now do the same with this list, and describe the shape. - I said E (previous question asks the same thing and I said it looks like a rhombus, which was right, this one is telling me to describe it [in other words, describe what a rhombus looks like] but I'm not exactly sure which triangle would best describe it)

(0,-1), (0,3), (1,1), (-1,1)
A A square
B Two acute angles that meet at the acute angles
CA triangle
D Two right triangles that meet at the right angles
E A rhombus
F Two obtuse triangles that meet at the hypotenuse

3. (-1, 0), (2, 0), (2, 4), and (-1, 4) - I said B (this question is asking what shape do these points make. I said square but I guess this is more of a RECTANGLE so I would say A)
A a rectangle
B a square
Ca parallelogram
D a rhombus
E a triangle
F a line segment

Bob · 2013-09-24 19:41:59

hi demha

1. Draw a Cartesian Coordinate system for yourself, and then determine whether these sets of points form triangles - I said F

Set 1: (0,0), (1,1), (0,1)
Set 2: (-1, 0), (1,1), (3,2)
Set 3: (0,0), (5,0), (3,5)
A set 1 line segment, set 2 Triangle, set 3 line segment
B set 1 Triangle, set 2 Triangle, set 3 line segment
C set 1 Triangle, set 2 line segment, set 3 Triangle
D set 1 line segment, set 2 line segment, set 3 line segment
E set 1 Triangle, set 2 line segment, set 3 line segment
F set 1 line segment, set 2 Triangle, set 3 Triangle

I could do a diagram, but I know you like to work these out for yourself. When I did this one, I accidentally put one point in the wrong place (swapped the x and y round) and that alters the question completely. So my advice is, do a new graph and check your points very carefully.

2. Now do the same with this list, and describe the shape. - I said E (previous question asks the same thing and I said it looks like a rhombus, which was right, this one is telling me to describe it [in other words, describe what a rhombus looks like] but I'm not exactly sure which triangle would best describe it)

(0,-1), (0,3), (1,1), (-1,1)
A A square
B Two acute angles that meet at the acute angles
CA triangle
D Two right triangles that meet at the right angles
E A rhombus
F Two obtuse triangles that meet at the hypotenuse

I can see why you chose a rhombus for this. But I think they intend you to join the points together in the order given. Try this and I think you'll see that another answer is then visible.

3. (-1, 0), (2, 0), (2, 4), and (-1, 4) - I said B (this question is asking what shape do these points make. I said square but I guess this is more of a RECTANGLE so I would say A)
A a rectangle
B a square
Ca parallelogram
D a rhombus
E a triangle
F a line segment

For a square all the sides must be equal. Your revised answer, A, is correct.

Bob

anonimnystefy · 2013-09-24 21:18:02

Hi Bob

demha · 2013-09-24 22:11:49

Hi Anonimnystefy,
It was really tempting to click on that spoiler, but I have resisted... for now.

Hi Bob,
I did #1 again.
First Set looks like a small triangle.
Second Set looks like a line segement.
Third Set looks like triangle.

My new answer would be C.

Bob · 2013-09-24 22:58:34

demha wrote:

My new answer would be C.

Yes, that's what I think too.

Stefy wrote:

hidden answer

Yes, I do. And thank you for the implied compliment.

Bob

demha · 2013-09-25 00:18:24

2. Now do the same with this list, and describe the shape. - I said E (previous question asks the same thing and I said it looks like a rhombus, which was right, this one is telling me to describe it [in other words, describe what a rhombus looks like] but I'm not exactly sure which triangle would best describe it)

(0,-1), (0,3), (1,1), (-1,1)
A A square
B Two acute angles that meet at the acute angles
CA triangle
D Two right triangles that meet at the right angles
E A rhombus
F Two obtuse triangles that meet at the hypotenuse

I would say for this one - F

By the way, my teacher replied for the previous math lesson and got 10.000!

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