MATH questions - It has begun once again!

[Orks da best]

I hate maths, I can do enough to get by but you all might as well be speaking Japanese for all the sense it makes to me.

I agree, the simple math is more importent then the op's in my mind, best be burned with a flamer...or two.

 

[isometry]

A man drops a ball off of a building. It takes 10 seconds to hit the ground. The ball weighs basically nothing and there is no wind, the ball generates negligable air resistance. How tall is the building.

Equation for a falling object near the earth's surface: g = 9.8 m/s²

So, 10 * 9.8 = 98 meters??

distance = 1/2 g t^2 , g = 10 m / s^2 so distance = 1/2(10)(10)^2 = 500.

Incidentally, to the others, the time it takes to fall "t = 10" is given to only one significant digit, so using g = 10 is more correct than g = 9.8. Students always waste their brain energy on "9.8", when it's so easy to do mental math with g = 10 and in this case is more correct.

 

[4RM3D]

*double post*

 

[Melon Hunter]

... yeah... next time, give us something proper by way of questions, like game theory or differential equations, or even geometry, ffs!

I started differential equations today.

I would've been all like "COME AT ME BRO!"

x''(t) + 4x'(t) + 4x = ?

Off you go!

That's not a differential equation, you have to put something on the RHS. You are supposed to figure out the unknown function x(t), not guess what the RHS is.

Sorry, I dun goofed. That question mark should be a zero.

Then as a general solution, x = (A+Bt)e[sup]-2t[/sup]

Excellent! I could go on into partial differential equations, but I think that's a little ridiculous to be throwing around, considering the questions asked so far.

 

[Redingold]

Here's a question that managed to stump my maths teacher today:

Use the substitution z = sin(x) to transform the equation cos(x)d[sup]2[/sup]y/dx[sup]2[/sup] + sin(x)dy/dx - 2ycos[sup]3[/sup](x) = 2cos[sup]5[/sup](x) into the equation
d[sup]2[/sup]y/dz[sup]2[/sup] - 2y = 2(1 - z[sup]2[/sup]), and hence solve the equation cos(x)d[sup]2[/sup]y/dx[sup]2[/sup] + sin(x)dy/dx - 2ycos[sup]3[/sup](x) = 2cos[sup]5[/sup](x), giving y in terms of x.

 

[SckizoBoy]

Here is a quickie from abusing Liebniz notation, although it can be rigorously justified.

Since y(x) = ln(x), x = e^y, so dx/dy = e^y, so treating this as a fraction dy/dx = 1/e^y = 1/x.

The fraction abuse is the statement that the derivative of the inverse function is the reciprocal of the derivative of the function, which is true in general (when the denominator is not zero) and can be derived by applying the chain rule to f^{-1}(f(x)) = x.

Oh, yeah, that old peach. Knew that one, but I always felt as though it was the 'easy way out', even though it patently provable that f'(x) = e[sup]x[/sup] when f(x) = e[sup]x[/sup] (hence my mention of the Taylor series). Still, this is starting to tax my meagre brain (too used to synth chem atm... -_- ).

Actually, thinking back, with the derivative of sin(x), you can do it by processing it's Taylor series... though somewhere niggling at the back of my mind, I think that it's actually done the other way around! (The derivatives prove the Taylor series...) *massive derp* !!

Equation for a falling object near the earth's surface: g = 9.8 m/s²

So, 10 * 9.8 = 98 meters??

Answer at top of page. And you've given speed as 9.8 as opposed to acceleration.

s = ut + (1/2)at[sup]2[/sup]

So s = 0 * 10 + (1/2) * 9.8 * 10[sup]2[/sup]

 

[4RM3D]

distance = 1/2 g t^2 , g = 10 m / s^2 so distance = 1/2(10)(10)^2 = 500.

Incidentally, to the others, the time it takes to fall "t = 10" is given to only one significant digit, so using g = 10 is more correct than g = 9.8. Students always waste their brain energy on "9.8", when it's so easy to do mental math with g = 10 and in this case is more correct.

Thanks for pointing that out. I'm sleeping* sleepy. Totally forgot about the square. Well, not that it mattered. My answer would still have been wrong... Less wrong, but wrong still.

EDIT: *(damn, that would have been awesome)

 

[SckizoBoy]

Here's a question that managed to stump my maths teacher today:

Use the substitution z = sin(x) to transform the equation cos(x)d[sup]2[/sup]y/dx[sup]2[/sup] + sin(x)dy/dx - 2ycos[sup]3[/sup](x) = 2cos[sup]5[/sup](x) into the equation
d[sup]2[/sup]y/dz[sup]2[/sup] - 2y = 2(1 - z[sup]2[/sup]), and hence solve the equation cos(x)d[sup]2[/sup]y/dx[sup]2[/sup] + sin(x)dy/dx - 2ycos[sup]3[/sup](x) = 2cos[sup]5[/sup](x), giving y in terms of x.

*meh* You're maths teacher's just too lazy to look at it properly!

Express dz/dx, insert and cancel away... (too lazy and tired to bother writing it out... -_- )

 

[BiscuitTrouser]

A man drops a ball off of a building. It takes 10 seconds to hit the ground. The ball weighs basically nothing and there is no wind, the ball generates negligable air resistance. How tall is the building.

Equation for a falling object near the earth's surface: g = 9.8 m/s²

So, 10 * 9.8 = 98 meters??

Nope, while its initial speed may be 9,8 gravity keeps pulling downward at 9.8 meters/sec/sec meaning the ball accelerates, it has no terminal velocity as it there is no air resistance. Applying your logic no object can fall faster than 9.8 m per second. Acceleration needs to be accounted for with a set of equations called SUVAT.

The ball falls for 10 seconds,

T = 10.
Acceleration = 9.8
Intial velocity (U) = 0 as he holds it.

S (distance moved) = TU + 1/2(A*(T^2))

This means that our distance = 0 + 1/2(9,8*100)

So it moved 490 meters all the way down. That means the building is 490 meters tall.

 

[SckizoBoy]

And now that I got that out of the way, here's a cool puzzle I heard a while ago:

Alice and Bob are playing a guessing game.
Alice: "I have three kids. Multiplying their ages together equals 36 and the sum of their ages equals the number of my neighbours house. Can you guess their ages?"
Bob goes and checks the house number, then comes back.
Bob: "I'll need more info."
Alice: "Alright. My oldest child plays piano."
Bob: "Okay, I got it now..."

...How old are the kids?

One nine-year-old and two two-year-olds... (based on integer-answers)...

 

[4RM3D]

Nope, while its initial speed may be 9,8 gravity keeps pulling downward at 9.8 meters/sec/sec meaning the ball accelerates, it has no terminal velocity as it there is no air resistance. Applying your logic no object can fall faster than 9.8 m per second. Acceleration needs to be accounted for with a set of equations called SUVAT.

The ball falls for 10 seconds,

T = 10.
Acceleration = 9.8
Intial velocity (U) = 0 as he holds it.

S (distance moved) = TU + 1/2(A*(T^2))

This means that our distance = 0 + 1/2(9,8*100)

So it moved 490 meters all the way down. That means the building is 490 meters tall.

*nods* *points to post*

Thanks for pointing that out. I'm sleeping* sleepy. Totally forgot about the square. Well, not that it mattered. My answer would still have been wrong... Less wrong, but wrong still.

EDIT: *(damn, that would have been awesome)

Time to get some coffee. I need to stop making stupid mistakes.

 

[isometry]

Here is a quickie from abusing Liebniz notation, although it can be rigorously justified.

Since y(x) = ln(x), x = e^y, so dx/dy = e^y, so treating this as a fraction dy/dx = 1/e^y = 1/x.

The fraction abuse is the statement that the derivative of the inverse function is the reciprocal of the derivative of the function, which is true in general (when the denominator is not zero) and can be derived by applying the chain rule to f^{-1}(f(x)) = x.

Oh, yeah, that old peach. Knew that one, but I always felt as though it was the 'easy way out', even though it patently provable that f'(x) = e[sup]x[/sup] when f(x) = e[sup]x[/sup] (hence my mention of the Taylor series). Still, this is starting to tax my meagre brain (too used to synth chem atm... -_- ).

Actually, thinking back, with the derivative of sin(x), you can do it by processing it's Taylor series... though somewhere niggling at the back of my mind, I think that it's actually done the other way around! (The derivatives prove the Taylor series...) *massive derp* !!

Yeah, I've seen analysis books where exp, sine, and cosine are defined by their Taylor series and properties like exp(A)exp(B) = exp(A+B) are derived by multiplying the infinite series, using the binomial expansion, etc. So if one wanted to be a wise-ass he could take the power series definition of sine as "first principles", and derive it's geometric properties and the sum-angle formulas from there.

Here's a question that managed to stump my maths teacher today:

Use the substitution z = sin(x) to transform the equation cos(x)d[sup]2[/sup]y/dx[sup]2[/sup] + sin(x)dy/dx - 2ycos[sup]3[/sup](x) = 2cos[sup]5[/sup](x) into the equation
d[sup]2[/sup]y/dz[sup]2[/sup] - 2y = 2(1 - z[sup]2[/sup]), and hence solve the equation cos(x)d[sup]2[/sup]y/dx[sup]2[/sup] + sin(x)dy/dx - 2ycos[sup]3[/sup](x) = 2cos[sup]5[/sup](x), giving y in terms of x.

That's pretty straightforward. Calculate the derivatives to be replaced using the chain rule:

y'(x) = y'(z)*z'(x) = y'(z) cos(x)

y''(x) = y''(z)*z'(x) + y'(z)*z''(x) = y''(z) cos(x) - y'(z) sin(x)

Plug these into the equation and collect the trig terms.

Edit: here is a small challenge problem. Calculate (0.99)^10 using only mental math, it should take just a few seconds.

 

[4RM3D]

One nine-year-old and two two-year-olds... (based on integer-answers)...

All the info you'll need is there and there's nothing unnecessary (except the framing).
The sum part is important.

9 + 2 + 2 ? That would mean the sum is 13. But what does that matter? Incidentally, 13 * 3 = 36. But I still don't see it.

 

[Scepta]

And now that I got that out of the way, here's a cool puzzle I heard a while ago:

Alice and Bob are playing a guessing game.
Alice: "I have three kids. Multiplying their ages together equals 36 and the sum of their ages equals the number of my neighbours house. Can you guess their ages?"
Bob goes and checks the house number, then comes back.
Bob: "I'll need more info."
Alice: "Alright. My oldest child plays piano."
Bob: "Okay, I got it now..."

...How old are the kids?

One nine-year-old and two two-year-olds... (based on integer-answers)...

Ok im really wondering how you got to that answer. I get the fact that the oldest kid is not part of a twin. I hate not being able to figure out riddles xD.

-edit Wait nevermind I understand! w00t. It's so annoyingly simple it sucks lol.

 

[Llil]

...How old are the kids?

One nine-year-old and two two-year-olds... (based on integer-answers)...

You are correct.

9 + 2 + 2 ? That would mean the sum is 13. But what does that matter? Incidentally, 13 * 3 = 36. But I still don't see it.

Bob knows the house number, so that would mean he knows the sum, right? That's important too.

---

Now how about the other question/task I mentioned briefly, the "real maths" -one.
"Prove that irrational numbers exist, or in other words, there exists a number that cannot be expressed as a fraction."

 

[isometry]

Now how about the other question/task I mentioned briefly, the "real maths" -one.
"Prove that irrational numbers exist, or in other words, there exists a number that cannot be expressed as a fraction."

The rational numbers are countable i.e. they can be mapped in one-to-one correspondence onto the natural numbers. Here's one construction:

1, 1/2, 2/2, 1/3, 2/3, 3/3,...

So the cardinality of the rational numbers is equal to the cardinality of the natural numbers. Intuitively the statement is "the rational numbers can be listed."

There can be no such correspondence for the real numbers, however, because they have a larger cardinality. Intuitively, this means the real numbers cannot be listed. For suppose you had a list L of all real numbers between 0 and 1:

0.a11 a12 a13 ...
0.a21 a22 a23 ...
0.a31 a32 a33 ...

etc, where aXY is the Yth decimal digit of the Xth real number in the list. Then we can construct a new real number between zero and 1 that is not on the list, contradicting the assumption. We construct this number by making the first digit anything other than a11, the second digit anything other than a22, and so on, so that in the end the number we've constructed differs from every number in the above list in at least one place. There for L is not a list of all real numbers between zero and one, which means there can be no such list.

Since the cardinality of the reals is greater than the cardinality of the rationals, there must be infinitely many real numbers that cannot be expressed as a fraction.

P.S. I didn't use the square root of 2 proof because I like analysis more than number theory.

 

[Auron225]

I get enough maths to do in uni to want to do more in my spare time o.o

 

[Llil]

P.S. I didn't use the square root of 2 proof because I like analysis more than number theory.

The square root proof was what I had in mind, and I think it's a bit more... elegant. But the one you posted is pretty neat too.

 

[NastoK]

Uuuu, maaaath. Right, just gonna leave this little comment here so I remember the thread. Hope that's okay D:

 

[4RM3D]

9 + 2 + 2 ? That would mean the sum is 13. But what does that matter? Incidentally, 13 * 3 = 36. But I still don't see it.

Bob knows the house number, so that would mean he knows the sum, right? That's important too.

Bob knows the house number, yes. But I do not. And I am suppose to imagine being Bob in that riddle. If I knew the neighbors house number was 13, then I would have known there would be only 1 solution (9, 2, 2). But how am I suppose to know the house number is 13?

Meh, it is probably so simple I can't even think of it. XD