Math Problems

I changed my mind. A whole brick length. 98 bricks are a half brick length offset from the first one, and the final one is a half brick length offset from those. The centre of gravity is, on average, a half brick length across, just at the very extreme of the stable range.

Edit: On second thoughts, there'd be more stable ways of constructing it, but I still think it'd be one brick length maximum to keep the centre of gravity from toppling it over.

I changed my mind. A whole brick length. 98 bricks are a half brick length offset from the first one, and the final one is a half brick length offset from those. The centre of gravity is, on average, a half brick length across, just at the very extreme of the stable range.

Edit: On second thoughts, there'd be more stable ways of constructing it, but I still think it'd be one brick length maximum to keep the centre of gravity from toppling it over.

Wouldn't that arrangement fall over?

Wouldn't that arrangement fall over?

In practice, yes. I was modelling the bricks as perfect rectangular prisms, the Earth's gravitational influence as a perfectly linear field and the accuracy of their stacking as being perfectly precise, as well as in isolation from any interfering effects such as wind - if my understanding of mechanics is correct, I think that such a system would be on the very edge of stability.

Quote from: Euclid on September 04, 2009, 09:57:29 PM

Wouldn't that arrangement fall over?

In practice, yes. I was modelling the bricks as perfect rectangular prisms, the Earth's gravitational influence as a perfectly linear field and the accuracy of their stacking as being perfectly precise, as well as in isolation from any interfering effects such as wind - if my understanding of mechanics is correct, I think that such a system would be on the very edge of stability.

Did you consider the torque on the upper 99 bricks?

I think it is >1.
could be wrong though.

I think it is >1.
could be wrong though.

Yeah.

Did you consider the torque on the upper 99 bricks?

I did, but I only just realised that I considered it incorrectly.

I suppose that's what I get for trying to do physics while feeling tired and lethargic.

Quote from: optimisticcynic on September 02, 2009, 07:47:43 AM

x+y=dy/dx. solve for y

dy/dx - y = x

m(x) = exp[int[-dx]] = exp(-x)

exp(-x) dy/dx - exp(-x) y = x exp(-x)

d[exp(-x) y]/dx = x exp(-x)

The integral is done by parts:

Int[x exp(-x) dx] = -x exp(-x) + int[exp(-x) dx] = -x exp(-x) - exp(-x) + C

u = x, dv = exp(-x) dx

du = dx, v = -exp(-x)

so we have

exp(-x) y = -x exp(-x) - exp(-x)  + C

y = -1 - x + C exp(x)

don't you need to put something to mention that exp is an e exponential. I normally right it out by hand so I don't know what is acceptable format for being typed.

Quote from: Euclid on September 04, 2009, 10:03:34 PM

Did you consider the torque on the upper 99 bricks?

I did, but I only just realised that I considered it incorrectly.

I suppose that's what I get for trying to do physics while feeling tired and lethargic.

golden rule of mathematics. never trust a proof you came up with after 11:00

don't you need to put something to mention that exp is an e exponential. I normally right it out by hand so I don't know what is acceptable format for being typed.

exp(x) is a commonly used and generally accepted way of typing e^x.

HAHA UR ALL GEEKS LOL 

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

1/2A*(1-A^2)^(1/2)=S for the area
the length of the boundary is 1+A+(1-A^2)^(1/2) right?

Quote from: Euclid on September 04, 2009, 08:18:55 PM

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

1/2A*(1-A^2)^(1/2)=S for the area
the length of the boundary is 1+A+(1-A^2)^(1/2) right?

huh?  "A" labels a point.  It's not a number.

Quote from: optimisticcynic on September 05, 2009, 07:13:19 AM

Quote from: Euclid on September 04, 2009, 08:18:55 PM

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

1/2A*(1-A^2)^(1/2)=S for the area
the length of the boundary is 1+A+(1-A^2)^(1/2) right?

huh?  "A" labels a point.  It's not a number.

I lol'd.

Anyway, I've been giving that problem some thought, and no luck yet. Granted, I started thinking about it while tired, then went to work and didn't have much time to dwell on it, and now am feeling tired again, so I wouldn't have expected results yet.

Quote from: Roundy the Truthinessist on September 04, 2009, 09:23:58 PM

Quote from: Euclid on September 04, 2009, 09:13:49 PM

Quote from: Roundy the Truthinessist on September 04, 2009, 09:09:02 PM

Quote from: Euclid on September 04, 2009, 08:29:29 PM

You have a set of one hundred bricks.  At the edge of the cliff, you want to stack the bricks on top of each other so that they extend as far as possible over the edge of the cliff.  What is the maximum length such an a structure can extend over the cliff (in terms of brick lengths)?

a bit more than 4?

No, less.

Ah, closer to 3 1/2 maybe?

This is based on the harmonic series, right?  It would be 1/2 + 1/2 + 1/3 + 1/4 ... 1/100?

So it's approximately ln 100 - the euler mascharoni constant - 1/2, or 4.605 - .577 -.5 = 3.528.

I know it's only an approximation but it's close, right?

Or am I totally wrong?

You're on the right track, but not right.

I think I have it now.

It's 1 + 1/2 + 1/4 + 1/6...+ 1/200.

1/2 ln 100 - 1/2(.577) - 1/2 = 2.302 - .289 - .5 = 1.5 bricklengths.  Approximately.

Quote from: optimisticcynic on September 05, 2009, 07:13:19 AM

Quote from: Euclid on September 04, 2009, 08:18:55 PM

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

1/2A*(1-A^2)^(1/2)=S for the area
the length of the boundary is 1+A+(1-A^2)^(1/2) right?

huh?  "A" labels a point.  It's not a number.

I meant As Y coordinate since it if I understand it correctly is (0,0<=A<=1) or a point on the y axis between 0 and 1. granted I could have misunderstood the question. I do that a lot.

Quote from: Euclid on September 05, 2009, 08:17:28 AM

Quote from: optimisticcynic on September 05, 2009, 07:13:19 AM

Quote from: Euclid on September 04, 2009, 08:18:55 PM

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

1/2A*(1-A^2)^(1/2)=S for the area
the length of the boundary is 1+A+(1-A^2)^(1/2) right?

huh?  "A" labels a point.  It's not a number.

I meant As Y coordinate since it if I understand it correctly is (0,0<=A<=1) or a point on the y axis between 0 and 1. granted I could have misunderstood the question. I do that a lot.

No, A is not a coordinate.

Quote from: optimisticcynic on September 05, 2009, 10:59:46 AM

Quote from: Euclid on September 05, 2009, 08:17:28 AM

Quote from: optimisticcynic on September 05, 2009, 07:13:19 AM

Quote from: Euclid on September 04, 2009, 08:18:55 PM

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

1/2A*(1-A^2)^(1/2)=S for the area
the length of the boundary is 1+A+(1-A^2)^(1/2) right?

huh?  "A" labels a point.  It's not a number.

I meant As Y coordinate since it if I understand it correctly is (0,0<=A<=1) or a point on the y axis between 0 and 1. granted I could have misunderstood the question. I do that a lot.

No, A is not a coordinate.

then I don't understand the question.

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

i might be completely wrong but isn't that just the definition of an triangle AB0 ( A=(0,a); B=(b,0) 0=(0,0))
area is 1/2ab, lenght of boundary a+b+sqrt(a^2+b^2) , segments of the boundary are y=0 for 0<=x<=b, x=0 for 0<=y<=a and y=a-(a/b)x for 0<=x<=b

What is wrong with you people? The question is very clearly stated and you're all horribly misinterpreting it.

Quote from: Euclid on September 04, 2009, 08:18:55 PM

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

i might be completely wrong but isn't that just the definition of an triangle AB0 ( A=(0,a); B=(b,0) 0=(0,0))
area is 1/2ab, lenght of boundary a+b+sqrt(a^2+b^2) , segments of the boundary are y=0 for 0<=x<=b, x=0 for 0<=y<=a and y=a-(a/b)x for 0<=x<=b

that's what I thought. I just defined B in terms of A since we know the distance from A to B(B=(b,0), A=(0,b),  b=sqrt(1-a^2). apparently I am miss interpreting it though

Quote from: Euclid on September 04, 2009, 09:26:47 PM

Quote from: Roundy the Truthinessist on September 04, 2009, 09:23:58 PM

Quote from: Euclid on September 04, 2009, 09:13:49 PM

Quote from: Roundy the Truthinessist on September 04, 2009, 09:09:02 PM

Quote from: Euclid on September 04, 2009, 08:29:29 PM

You have a set of one hundred bricks.  At the edge of the cliff, you want to stack the bricks on top of each other so that they extend as far as possible over the edge of the cliff.  What is the maximum length such an a structure can extend over the cliff (in terms of brick lengths)?

a bit more than 4?

No, less.

Ah, closer to 3 1/2 maybe?

This is based on the harmonic series, right?  It would be 1/2 + 1/2 + 1/3 + 1/4 ... 1/100?

So it's approximately ln 100 - the euler mascharoni constant - 1/2, or 4.605 - .577 -.5 = 3.528.

I know it's only an approximation but it's close, right?

Or am I totally wrong?

You're on the right track, but not right.

I think I have it now.

It's 1 + 1/2 + 1/4 + 1/6...+ 1/200.

1/2 ln 100 - 1/2(.577) - 1/2 = 2.302 - .289 - .5 = 1.5 bricklengths.  Approximately.

Very nearly right except for a small error.

Quote from: iznih on September 05, 2009, 05:10:56 PM

Quote from: Euclid on September 04, 2009, 08:18:55 PM

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

i might be completely wrong but isn't that just the definition of an triangle AB0 ( A=(0,a); B=(b,0) 0=(0,0))
area is 1/2ab, lenght of boundary a+b+sqrt(a^2+b^2) , segments of the boundary are y=0 for 0<=x<=b, x=0 for 0<=y<=a and y=a-(a/b)x for 0<=x<=b

that's what I thought. I just defined B in terms of A since we know the distance from A to B(B=(b,0), A=(0,b),  b=sqrt(1-a^2). apparently I am miss interpreting it though

Yeah, you're not understanding it well.  Maybe a visual example will help.  Say you have a ladder set against a wall.  It can be set at a variety of angles to the wall.  If a point falls within a right triangle formed by the ladder at some possible angle, then it is a member of the region I refer to.

ah,i think i get it. one boundary is a quarter of a circle

ah,i think i get it. one boundary is a quarter of a circle

No.

Quote from: optimisticcynic on September 05, 2009, 06:01:35 PM

Quote from: iznih on September 05, 2009, 05:10:56 PM

Quote from: Euclid on September 04, 2009, 08:18:55 PM

A line segment AB of length 1 has endpoints A, constrained to the nonnegative y axis, and B, constrained to the nonnegative x axis.  Let the region S of the xy plane be defined as follows.  A point P is in S if P lies within a right triangle formed by A, B, and the origin for some A,B satisfying the condition above.

What is the area of S?  What is the length of its boundary?  Find an equation of each smooth segment of the boundary of S in Cartesian coordinates.

i might be completely wrong but isn't that just the definition of an triangle AB0 ( A=(0,a); B=(b,0) 0=(0,0))
area is 1/2ab, lenght of boundary a+b+sqrt(a^2+b^2) , segments of the boundary are y=0 for 0<=x<=b, x=0 for 0<=y<=a and y=a-(a/b)x for 0<=x<=b

that's what I thought. I just defined B in terms of A since we know the distance from A to B(B=(b,0), A=(0,b),  b=sqrt(1-a^2). apparently I am miss interpreting it though

Yeah, you're not understanding it well.  Maybe a visual example will help.  Say you have a ladder set against a wall.  It can be set at a variety of angles to the wall.  If a point falls within a right triangle formed by the ladder at some possible angle, then it is a member of the region I refer to.

Ok I get it

Quote from: optimisticcynic on September 04, 2009, 10:15:09 PM

I think it is >1.
could be wrong though.

Yeah.

Yeah I am wrong or yeah the answer is >1

ah,i think i get it. one boundary is a quarter of a circle

sometimes i'm ashamed of myself  
it's clearly not a circle

edit: typo

Quote from: iznih on September 05, 2009, 06:25:20 PM

ah,i think i get it. one boundary is a quarter of a circle

someties i'm ashamed of myself 
it's clearly not a circle

Could you please spare us from your mental monologues and give a final solution if you can. If not, don't post. Also, we are not here to solve homework problems, since this problem is clearly taken from a Calculus textbook.

Quote from: iznih on September 06, 2009, 09:12:20 AM

Quote from: iznih on September 05, 2009, 06:25:20 PM

ah,i think i get it. one boundary is a quarter of a circle

someties i'm ashamed of myself 
it's clearly not a circle

Could you please spare us from your mental monologues and give a final solution if you can. If not, don't post. Also, we are not here to solve homework problems, since this problem is clearly taken from a Calculus textbook.

Now you know.