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Maths Fun(?) #873397
03/12/13 02:18 PM
03/12/13 02:18 PM
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Bucks, England
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-1. Draw a regular hexagon then draw the six "alternate" vertices. (Skip one vertex - 1 to 3, 2 to 4 etc). If the area of the original hexagon is 6, what is the area of the internal hexagon that is formed?

-2. Inscribe a square in a circle of radius R. Draw an arc from one corner of the square with radius equal to the side of the square. The area formed between this arc and the circle is a crescent. Find the area of this crescent in terms of R.

-3. In the equations below, replace each letter with a number from 1 to 9, using each number only once:

a/b + c/12 = 1 1/12
d/e + 5/f = 7/8
g/h + 3/i = 1 1/2


To waste one second of one's life is a betrayal of one's self.
I wonder what's on television...?
Re: Maths Fun(?) [Re: Urban Worrier] #873457
03/12/13 04:38 PM
03/12/13 04:38 PM
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Unionville
manxman Offline
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2. 1\2*R^2 wave


Sometimes lost is where you need to be. Just because you don't know your direction doesn't mean you don't have one.
Re: Maths Fun(?) [Re: Urban Worrier] #873480
03/12/13 06:28 PM
03/12/13 06:28 PM
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Mojave desert, California
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3.

2/3 + 5/12 = 1 1/12
1/4 + 5/8 = 7/8
7/6 + 3/9 = 1 1/2

smile

Re: Maths Fun(?) [Re: Urban Worrier] #873586
03/13/13 11:21 AM
03/13/13 11:21 AM
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Posts: 18,719
Ottawa Ontario Canada
CanukDenis Offline
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1. Draw a regular hexagon then draw the six "alternate" vertices.
(Skip one vertex - 1 to 3, 2 to 4 etc). If the area of the original hexagon is 6,
what is the area of the internal hexagon that is formed?

So I don't have to type "sqrt(3)"!):
u = sqrt(3)
v = 1/u (ratio radii outer:inner)

R = outer radius
Pentagon area: 3u(R^2) / 2

Given: area = 6 ; soooooo:
3u(R^2) / 2 = 6
R = sqrt(4 / u) : ~1.5196....

r = inner radius
r = R * v : ~.8773....
area = 3u(r^2)/2
area = (drum rolls!) 2 exactement.








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Re: Maths Fun(?) [Re: Urban Worrier] #873654
03/13/13 06:30 PM
03/13/13 06:30 PM
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Posts: 21,813
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CCb - very nice!

Denis - good solve. It can be done by similar triangles. The inner hexagon contains 6 identically-sized equilateral triangles and the outer hexagon contains these 6 plus 6 more plus 12 halves for a total of 18 triangles. Thus, the area of the outer hexagon is 3 times the area of the inner one. smile

manxman - if I read your answer correcly (one over 2 R-squared) then we beg to differ. What's your working? think


To waste one second of one's life is a betrayal of one's self.
I wonder what's on television...?
Re: Maths Fun(?) [Re: Urban Worrier] #873701
03/13/13 11:45 PM
03/13/13 11:45 PM
Joined: Jul 2002
Posts: 29,026
Unionville
manxman Offline
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Sorry, checked my work over and got R^2 smile

Last edited by manxman; 03/14/13 07:32 AM.

Sometimes lost is where you need to be. Just because you don't know your direction doesn't mean you don't have one.
Re: Maths Fun(?) [Re: Urban Worrier] #873814
03/14/13 01:26 PM
03/14/13 01:26 PM
Joined: Nov 2004
Posts: 21,813
Bucks, England
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Great solve! grin


To waste one second of one's life is a betrayal of one's self.
I wonder what's on television...?
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