If I remember right the volume of an octahedron of edge length l is:

v=1/3 Sqrt(2) l^3

which is : 1.414213562 * 1/3* 1 =.4714... cubic units

Correct again Bill! Your still in this Fred!

Q84 : 0.471404

Standings:
1. William Rothamel - 24 1/2 Points
2. Fred W - 21 Points
3. Guido.Man - 10 Points
4. Mr. Kevvy - 7 Points
5. Dominique - 6 1/2 Points
6. Philadelphia - 6 1/2 Points
7. WinterKnight - 6 Points
8. Sarge - 5 1/2 Points
9. Jason gee - 4 1/2 Points
10. John McLeod VII - 1 Point
11. Labbie - 1 Point
12. Scary Capitalist - 1/2 Point
13. TBD...

Question 85 (1 Point) : What is the volume of an tetrahedron with edge length of 1?

Luke.

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- Luke.

Height of one of the triangles (not the tetrahedron): sqrt(1^2-(1/2)^2) = 1/2*sqrt(3)

Area of one of the triangles: (1/2)*1*(1/2*sqrt(3)) = 1/4*sqrt(3)

Volume of the tetrahedron: (1/3) * Area of triangular base * Height of tetrahedron = (1/3) * (1/4*sqrt(3)) * Height of tetrahedron

Equilateral triangle, so altitudes are the same as medians.
Medians are concurrent.
Split 2 by 1.

Height of tetrahedron: sqrt((1/2*sqrt(3))^2-((1/3)*(1/2*sqrt(3)))^2) = 1/3*sqrt(6)

Volume = (1/3) * (1/4*sqrt(3)) * 1/3*sqrt(6) = (1/36) sqrt(18) = sqrt(2) / 12, which is approximately 0.1178511302.

(This is, of course, under the assumption that Luke meant a regular tetrahedron.)

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Capitalize on this good fortune, one word can bring you round ... changes.

Well done Sarge! Looks like he might be back in the game! Welcome back!

Q85 Answer: 0.1178

Standings:
1. William Rothamel - 24 1/2 Points
2. Fred W - 21 Points
3. Guido.Man - 10 Points
4. Mr. Kevvy - 7 Points
5. Dominique - 6 1/2 Points
6. Philadelphia - 6 1/2 Points
7. Sarge - 6 1/2 Points
8. WinterKnight - 6 Points
9. Jason gee - 4 1/2 Points
10. John McLeod VII - 1 Point
11. Labbie - 1 Point
12. Scary Capitalist - 1/2 Point
13. TBD...

Another Question - nearing the end...

Q86 (1 Point) : You are a cook in a remote area with no clocks or other way of keeping time other than a 4 minute hourglass and a 7 minute hourglass. You do have a stove however with water in a pot already boiling. Somebody asks you for a 9 minute egg, and you know this person is a perfectionist and will be able to tell if you undercook or overcook the eggs by even a few seconds. What is the least amount of time it will take to prepare the egg?

Luke.

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- Luke.

Well done Sarge! Looks like he might be back in the game! Welcome back!

Q85 Answer: 0.1178

Standings:
1. William Rothamel - 24 1/2 Points
2. Fred W - 21 Points
3. Guido.Man - 10 Points
4. Mr. Kevvy - 7 Points
5. Dominique - 6 1/2 Points
6. Philadelphia - 6 1/2 Points
7. Sarge - 6 1/2 Points
8. WinterKnight - 6 Points
9. Jason gee - 4 1/2 Points
10. John McLeod VII - 1 Point
11. Labbie - 1 Point
12. Scary Capitalist - 1/2 Point
13. TBD...

Another Question - nearing the end...

Q86 (1 Point) : You are a cook in a remote area with no clocks or other way of keeping time other than a 4 minute hourglass and a 7 minute hourglass. You do have a stove however with water in a pot already boiling. Somebody asks you for a 9 minute egg, and you know this person is a perfectionist and will be able to tell if you undercook or overcook the eggs by even a few seconds. What is the least amount of time it will take to prepare the egg?

Luke.

Q86: 16 minutes

Start both hourglasses at the same time and turn the 4 minute timer when it finishes. Put the egg in when the 7 minute timer finishes. Turn the 4 minute timer twice more, at 8 and 12 minutes and when it finishes again the egg is done.

F.

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Q-86

You should look at this just like the fuse problem. remember that one can time the other and you can get a partial load of sand (partial time)

start both hour glasses.put the egg in
restart 4 minute one when it finishes
restart 7 minute one when it finishes
When the 4 minute one finishes turn the 7 minute one over --it will have just 1 minute of sand
When the minute runs out of the 7 minute one you are done. Take the egg out

Exactly 9 minutes

Q-86

You should look at this just like the fuse problem. remember that one can time the other and you can get a partial load of sand (partial time)

start both hour glasses.put the egg in
restart 4 minute one when it finishes
restart 7 minute one when it finishes
When the 4 minute one finishes turn the 7 minute one over --it will have just 1 minute of sand
When the minute runs out of the 7 minute one you are done. Take the egg out

Exactly 9 minutes

B****r!!!

30-love.

F.

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Sorry Fred! Bill wins with the correct answer again! I'll give you one half-point anyway! Well Done to Bill! Your still in this Fred!

Standings:
1. William Rothamel - 25 1/2 Points
2. Fred W - 21 1/2 Points
3. Guido.Man - 10 Points
4. Mr. Kevvy - 7 Points
5. Dominique - 6 1/2 Points
6. Philadelphia - 6 1/2 Points
7. Sarge - 6 1/2 Points
8. WinterKnight - 6 Points
9. Jason gee - 4 1/2 Points
10. John McLeod VII - 1 Point
11. Labbie - 1 Point
12. Scary Capitalist - 1/2 Point
13. TBD...

Q87 : You are in a race in which the starting line is at a certain point on a straight beach. The finish line is in the water. One way to arrive at the finish line is to run 4 kilometers down the beach, make a 90 degree turn and swim 1 kilometer. However, you may cut into the water at any point. You speed on land is 6 k.p.h and you speed in water is 2 k.p.h.. At what point, measured from the starting line, should you cut into the water?

Luke.

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- Luke.

I assume the finish line is in the water ??

Bill

I assume the finish line is in the water ??

Yes I see that is in the question --DUH !

I will leave this for others should be a simple differentiation on a triangle for the minimum travel time.

Bill

Bill's takin' a break for this one... so anymore answers?

Q87 : You are in a race in which the starting line is at a certain point on a straight beach. The finish line is in the water. One way to arrive at the finish line is to run 4 kilometers down the beach, make a 90 degree turn and swim 1 kilometer. However, you may cut into the water at any point. You speed on land is 6 k.p.h and you speed in water is 2 k.p.h.. At what point, measured from the starting line, should you cut into the water?

Luke.

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- Luke.

For ease of expression, x is not the distance from the starting point to the place where the racer cuts into the water. It is the distance from the point where the trip would be along perpendicular segments back to the cutting point.

f := ((4-x)/6) + sqrt(1+x^2) = 2/3-1/6*x+1/2*sqrt(1+x^2)

f[prime] := -1/6+1/2/(1+x^2)^(1/2)*x

-1/6+1/2/(1+x^2)^(1/2)*x = 0

1/2/(1+x^2)^(1/2)*x = 1/6

3*x = sqrt(1+x^2)

9*x^2 = 1+x^2

8*x^2 = 1

x^2 = 1/8

(Obviously, a negative answer for x is undesired.)

x = 1/4*sqrt(2)

Thus, the answer is 4-1/4*sqrt(2), or approximately 3.646446610 ... .

[EDIT: Note f[prime,prime] = -1/2/(1+x^2)^(3/2)*x^2+1/2/(1+x^2)^(1/2). Substituting x results in 8/27*sqrt(2) > 0, indicating x really minimizes the function f.]

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Capitalize on this good fortune, one word can bring you round ... changes.

Now for something a bit different, I want one of you to mark Sarge's answer! Is it correct? Was the solution right? (I know the answer, but I want others of you to figure it out) 1 Point is up for grabs!

Luke.

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- Luke.

Ok, no one's up for it. So I'll answer Sarge by saying it's correct! Well Done Sarge!

Standings: (10 Posts to go!)
1. William Rothamel - 25 1/2 Points
2. Fred W - 21 1/2 Points
3. Guido.Man - 10 Points
4. Sarge - 7 1/2 Points
5. Mr. Kevvy - 7 Points
6. Dominique - 6 1/2 Points
7. WinterKnight - 6 Points
8. Jason gee - 4 1/2 Points
9. John McLeod VII - 1 Point
10. Labbie - 1 Point
11. Scary Capitalist - 1/2 Point
12. TBD...

And with 1 Point, Sarge Rockets himself into 4th place...

Q88 : One one side of a river are three humans, one big monkey, two small monkeys, and one boat. Each of the humans and the big monkey are strong enough to row the boat. The boat can fit one or two bodies (regardless of size). If at any time at either side of the river the monkeys outnumber the humans the monkeys will eat the humans. How do you get everyone on the other side of the river alive?

Luke.

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- Luke.

Ok, no one's up for it. So I'll answer Sarge by saying it's correct! Well Done Sarge!

Standings: (10 Posts to go!)
1. William Rothamel - 25 1/2 Points
2. Fred W - 21 1/2 Points
3. Guido.Man - 10 Points
4. Sarge - 7 1/2 Points
5. Mr. Kevvy - 7 Points
6. Dominique - 6 1/2 Points
7. WinterKnight - 6 Points
8. Jason gee - 4 1/2 Points
9. John McLeod VII - 1 Point
10. Labbie - 1 Point
11. Scary Capitalist - 1/2 Point
12. TBD...

And with 1 Point, Sarge Rockets himself into 4th place...

Q88 : One one side of a river are three humans, one big monkey, two small monkeys, and one boat. Each of the humans and the big monkey are strong enough to row the boat. The boat can fit one or two bodies (regardless of size). If at any time at either side of the river the monkeys outnumber the humans the monkeys will eat the humans. How do you get everyone on the other side of the river alive?

Luke.

Almost cracked Q88. Hoping to get there before Bill's insomnia kicks in again ;)

F.

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My answer for Q88: I've combined an outbound and return journey at each step to try make it easier to follow.

Start: 3 Humans, 1 Big monkey & 2 small monkeys on near side

1) Big monkey takes small monkey across, Big Monkey returns alone.
(Near=3 Humans, one small & One big monkey, far side = 1 small monkey)

2) Big monkey takes another small monkey across, & returns alone again.
(Near=3 Humans, & One big monkey, far side = 2 small monkeys)

3) Human takes a human across, returning with a small monkey.
(Near=2 Humans, One big monkey & one small, far side = 1 small monkey & 1 human)

4)Human takes a big monkey across, human returning with a small monkey.
(Near=2 Humans, & two small monkeys, far side = 1 big monkey & 1 human)

5) Human takes a Human across, Big Monkey returns alone.
(Near=2 small monkeys & 1 big monkey, far side = 3 humans)

6) Big monkey takes small monkey across, Big Monkey returns alone.
(Near=1 small monkey & 1 big, far side = 3 humans & 1 monkey)

7) Big monkey takes small monkey across,
(Near= none, far side = 3 humans & 2 small monkeys & 1 big)

Jason

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"Living by the wisdom of computer science doesn't sound so bad after all. And unlike most advice, it's backed up by proofs." -- Algorithms to live by: The computer science of human decisions.

Just woke up,

Answer to 88 for me

You can't do it at all --the monkeys eat the human before he can start with the boat

Congrats, Jason.

You got it while I was preparing the family lunch.

F.

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Just woke up,

Answer to 88 for me

You can't do it at all --the monkeys eat the human before he can start with the boat

I guess I like to work on the wrong problem all the time --I like to jump to the solution before I have finished reading the problem.

mmmmm, monkey brains ... ;P

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"Living by the wisdom of computer science doesn't sound so bad after all. And unlike most advice, it's backed up by proofs." -- Algorithms to live by: The computer science of human decisions.