Problems

Math Tricks/Problems

21:39:57 Oct 8th 07 - Mr. Asystole:

x^4 + (x+1)^4 = y^2 + (y+1)^2
x^2 + (x+1)^2 = y + (y+1)
2x^2 + 2x + 1 = 2y + 1
x^2 + x + 1/2 = y + 1/2
x^2 + x = y
x(x+1) = y
x = 0
x = -1 (negative)

21:42:00 Oct 8th 07 - Mr. Asystole:

Solutions are {x=0, y=1} and {x=-1, y=1}

01:29:44 Oct 9th 07 - Mr. Sigheart:

I got a different answer while I'm at it I might as well mention that you should Find all solutions in nonnegative integers to x⁴ + (x+1)⁴ = y² + (y+1)².

16:04:07 Oct 9th 07 - Mr. Sun:

x^4 + (x+1)^4 = y^2 + (y+1)^2
x^2 + (x+1)^2 = y + (y+1)

is this step right?

16:46:03 Oct 9th 07 - Mr. Sun:

(0,0) is one obvious answer. all thats left is to prove if thats the only solution, or show otherwise... :o

21:00:03 Oct 9th 07 - Mr. Sigheart:

Good job Sun you got it.

Here's my work

(a) y² + (y+1)² is strictly increasing as a function of y for nonnegative y.

x⁴ + (x+1)⁴ > (x²+x)² + (x²+x +1)² for x > 0; (difference is 2x²+2x)

Using result (a),

(1) y > x²+x for x > 0.

x⁴ + (x+1)⁴ < (x²+x+1)² + (x²+x+1 +1)² for x > 0; (difference is 2x²+2x+4)

Using result (a),

(2) y < x²+x+1 for x > 0.

Combining results (1) and (2) shows us that there are no solutions for x > 0.

Choosing x = 0 gives y = 0 as the only nonnegative solution.

21:20:47 Oct 9th 07 - Mr. Asystole:

Oops I meant (0,0). Substituted wrong there :P. Not sure if my methodology was right but curiously it returned the same answer.

02:41:18 Oct 10th 07 - Mr. Sigheart:

Mistakes happens to best of us.

A certain lake has an irregular shape. In order to estimate the area, the North-South distance across the lake is measured every 100 feet down an East-West line. The results are shown in the table below. Use an appropriate technique to estimate the number of acres that the lake covers. Give a complete explanation of how you computed the area.

East-West position (ft) Distance across lake (ft)

0 900

100 1500

200 1800

300 2000

400 2500

500 2700

600 2700

700 2700

800 2700

900 2700

1000 2700

1100 2700

1200 2700

1300 2700

1400 2600

1500 2600

1600 2600

1700 2600

1800 2600

1900 2600

2000 2500

2100 2500

2200 2500

2300 2500

2400 2500

2500 2500

2600 2400

2700 2300

2800 2300

2900 1800

3000 1600

3100 1500

3200 1400

3300 1400

3400 1300

3500 1300

3600 1100

3700 800

3800 0

02:53:09 Oct 10th 07 - Duke Tiber Septim IV:

I'm probably wrong but...189.40 acres?

04:00:12 Oct 10th 07 - Mr. Asystole:

A really rough way would be to just average the distances (sum them, and divide by the 38 intervals). Then multiply by 3800 (distance east-west). Then convert that to acres.

Creating a picture with some assumptions might help. The lake could be assymetric or very irregular, so any answer will be just an estimate.

Might also be able to use a polynomial or some sort of curve fitting.

05:23:01 Oct 10th 07 - Mr. Sigheart:

You could use the Trapezoidal Rule or the Simpson Rule and you'll get in about the same decimal placement like Septim did. While I'm at it Septim got it right.

07:22:34 Oct 10th 07 - Duke Tiber Septim IV:

OMG! I DID!?!

*Septim pulls a rope and a bunch of party decorations fall from the ceiling followed by a banner saying, "SEPTIM'S RIGHT!!!"

"Wasn't there supposed to be a clown?"

*A skeleton clown falls from the ceiling.

15:13:38 Oct 10th 07 - Mr. Sun:

<--- cannot even measure acres, only used to meters etc..

17:22:30 Oct 10th 07 - Mr. Verll:

1+1=4

Right?

20:41:06 Oct 10th 07 - Mr. Binh The Crazy:

right, it makes sense too.

one man marry one lady, they have two kids. Thus 1+1 = 4. :)

Anyway, haven't used trapezoidal rule since 5 years. :)

20:43:47 Oct 10th 07 - Mr. Sun:

never knew it was called the trapezoidal rule, i just know it as part of calculus lol

20:48:43 Oct 10th 07 - Mr. Acelnorst:

can someone explain me how 2+2=5 and 2+2=3 can both be true?

(This IS a math trick. Really old, actually)

21:00:17 Oct 10th 07 - Mr. Verll:

well there was two couples sleeping with each other. One of the chicks got a kid thus, 2+2=5 and the other one was that the guy in one of the relationships drugged his girlfriend, and when she was drugged gang banged her with 2 of his friends(one of them a woman), then the girlfriend woke up the day after, hurting in her (you know....) and then she made the guy tell him what he had done (NEVER MESS WITH A ANGRY WOMAN!!!!!) and she killed him out of rage, thus
2+2=3

simple

22:04:28 Oct 10th 07 - Duke Tiber Septim IV:

VERLL'S A GENIUS!!!

23:26:26 Oct 10th 07 - Mr. Atreides:

I don't know about that, but I know for a fact that 1=2. Prove me wrong? (it's not that hard....just look more carefully if you don't see it at first).

a=b=1
a=b
a^2=ab
a^2-b^2=ab-b^2
(a+b)(a-b)=b(a-b)
a+b=b
1+1=1
2=1

23:33:37 Oct 10th 07 - Mr. Verll:

Thanks :)

23:40:41 Oct 10th 07 - Mr. Spankylicious:

"a=b=1
a=b
a^2=ab
a^2-b^2=ab-b^2
(a+b)(a-b)=b(a-b)
a+b=b
1+1=1
2=1"

Very good! Now try it without simplifying with 0 and manage that. Last time I checked division by 0 wasn't a valid operation.

23:52:20 Oct 10th 07 - Mr. Atreides:

Wow...got that in less than 15 mins.......should've let others have a chance lol ;)

11:53:29 Oct 11th 07 - Mr. Oamawt:

Calculus sucks!

12:08:22 Oct 11th 07 - Mr. Frostmourne:

I second Oamawt's post

14:54:44 Oct 11th 07 - Mr. Spankylicious:

Calculus? I think you don't know what calculus really is. Everything here is 7th or 8th grade algebra.

15:43:03 Oct 11th 07 - Mr. Dreadii:

true how can you have calculis withouth inegrals and diffirentails lol.

15:46:44 Oct 11th 07 - Mr. Sun:

i highly doubt those are 7th-8th grade algebra for north american public schools. but can be that 7th-8th of many other places in the world.

15:52:30 Oct 11th 07 - Duke Tiber Septim IV:

*Septim feels sad

"..."

16:58:35 Oct 11th 07 - Mr. Verll:

*looks at Septim

"Whats the matter?"

17:34:09 Oct 11th 07 - Mr. Spankylicious:

"i highly doubt those are 7th-8th grade algebra for north american public schools. but can be that 7th-8th of many other places in the world."

Well I don't know what 7th-8th grade algebra is in the US, but that's what it is where I live.

17:39:25 Oct 11th 07 - Mr. Sun:

something like (1/4)*x+3=(3/4) i believe....

21:36:36 Oct 11th 07 - Mr. Sigheart:

If 4^x + 4^-x = 7, then what is 8^x + 8-x

thats not 7th or 8th anywhere. Neither is this x⁴ + (x+1)⁴ = y² + (y+1)². I feel insulted actually most of these are 9 through 12th grade problems, a few are 8th grade though. Besides you guys are fools if you didn't know you use Algebra throughout the rest of Highschool. In fact Spankylicous you've said a couple are 10th grade level.

But Spanky I'll give you a calculus one just for you.

A bowl is shaped like a hemisphere with diameter 30cm. A ball with diameter 10cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl.

05:17:13 Oct 12th 07 - Mr. Atreides:

Isn't that impossible to solve.......I haven't taken calculus, but still.....I don't see how that could be solved.

05:41:37 Oct 12th 07 - Mr. Sigheart:

A bowl is shaped like a hemisphere with diameter 30cm. A ball with diameter 10cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl.

Partially Submerged 0_<_h<10

Fully Submerged 10_<_h<15

Find the volume of the larger cap (r = 15)

Find the volume of the smaller cap (r = 5)

Just find the equation please.

The things with the lines right beside them mean that I am saying greater then or equal to.

06:30:03 Oct 12th 07 - Mr. Asystole:

Volume of a sphere = 4 / 3 * pi * r^3

V of bowl = 4 / 3 * pi * 15^3 / 2 (hemi)

V of ball = 4 / 3 * pi * 5^3

This is like Grade 12 Calc or so.....if I looked it up in a text I could get this. Something to do with solving an equation relating to volume of a sphere in terms of h and taking a differential I think, to get the water's volume.

15:40:18 Oct 12th 07 - Mr. Sun:

TRIPLE INTEGRAL YAHOOOO!

but you can use calculus, perhaps polar coordinates would be easier in this case to derive the volume formulas for spheres.

21:19:32 Oct 12th 07 - Mr. Dreadii:

volume formula ? evrybody has that memorised :D

21:31:05 Oct 12th 07 - Mr. Sigheart:

Then answer it and it'll be wrong.

21:52:12 Oct 12th 07 - Mr. Sun:

orly? -_-

11:10:37 Oct 13th 07 - Mr. Dreadii:

never had to do that kind of question.
i could look up the formula for a partial volume of a sphere.

Or are you suppose to come up with it yourself?

20:13:22 Oct 13th 07 - Mr. Sigheart:

Sure you can look it up.

00:18:51 Oct 14th 07 - Mr. Sun:

well, you can come up with it yourself dreadii.

for different h, you can find the crosssectional area (difference between big semi circle and cross area of the submerged part of the ball - arctan has to be used to get the angle? or maybe some simpler method)
then use it to integrate over 180 degree in polar coordinates, think of it like a fan spinning 180 degree.

18:44:11 Oct 15th 07 - Mr. Dragonfly:

fan swinging in 3D??
does it have to make endless circles??
never seen it could you show in an example :D

18:57:59 Oct 15th 07 - Mr. Dragonfly:

ooh i made a drawing.

integral S d. angle pi * (sin(angle))^2

= -cos (angle)* pi/3 * (Sin(angle))^3

is that correct?

19:05:49 Oct 15th 07 - Mr. Dragonfly:

btw i did it sidewards

19:15:04 Oct 15th 07 - Mr. Dragonfly:

R0 = 5

S = 0.5 * -cos (angle)* pi/3 * (Sin(angle))^3
(correction multiplied by 0.5)

h = 5 * cos (angle)

S from 0 to (cos^-1(h/5)) of 0.5 * -cos (angle)* pi/3 * (Sin(angle))^3

is the submerged volume of the ball.
Volume water is 2/3 * pi *15^3 Minus the Vol of bal above :D

thats partialy submerged the rest is ez.

correct me if im wrong pls.
BTW im Dreadii :p

19:25:02 Oct 15th 07 - Mr. Dragonfly:

simplefied V ball = - h * pi /15 * (sin (h/5)) ^3

19:31:33 Oct 15th 07 - Mr. Dragonfly:

i made a mistake again i post too fast give me i min :D

19:39:23 Oct 15th 07 - Mr. Dragonfly:

i got the definition of h all wrong :D

h = 5- cos(angle)*5
angle = cos^-1((5-h)/5)

V = 0.5 * -cos (cos^-1((5-h)/5))* pi/3 * (Sin(cos^-1((5-h)/5)))^3

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