anonymous
  • anonymous
hey @parthkohli
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
when i say 30 i mean 40
ParthKohli
  • ParthKohli
what does "reducing the length of the curve" mean? I mean you can draw the shortest possible path from (10,10) to (30,0) that also happens to be a straight line.
anonymous
  • anonymous
yea but it needs to be smooth

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anonymous
  • anonymous
so polynomial
ParthKohli
  • ParthKohli
what does that mean? a linear function is a polynomial too. what exactly do you mean by smooth?
anonymous
  • anonymous
mena no rough edges and its curve
anonymous
  • anonymous
like continuous function
ParthKohli
  • ParthKohli
ahhh, better.
anonymous
  • anonymous
is there function that looks like the black line i drew
ParthKohli
  • ParthKohli
so you need a polynomial function that makes a smooth transition and is also the shortest
anonymous
  • anonymous
so it has curve to smothly attach to another track but at the smae shorter length
anonymous
  • anonymous
yes
anonymous
  • anonymous
but at the same time *
ParthKohli
  • ParthKohli
what's the context of this question? it's really specific
anonymous
  • anonymous
i have done the question let me send the link
anonymous
  • anonymous
but what i am trying to figure out is the shorter length than this
ParthKohli
  • ParthKohli
sorry, my OS stopped working.
anonymous
  • anonymous
so on xy plane i want a function that give me length <27.37
anonymous
  • anonymous
|dw:1443516744812:dw|
anonymous
  • anonymous
ok
ParthKohli
  • ParthKohli
You're permitted to use calculators right?\[g(x) = ax^4 + bx^3 + cx^2 + dx + e\]\[10^4 a + 10^3 b + 10^2 c + 10 d + e = 10\]\[30^4 a + 30^3 b + 30^2 c + 30 d + e=0\]\[\int_{10}^{30} g(x)dx=200\]\[4a(10)^3 + 3b(10)^2 + 2c(10) + d(10) =0\]Now the last one is tricky.
anonymous
  • anonymous
yess
ParthKohli
  • ParthKohli
You know how to calculate arc length using integrals right?
anonymous
  • anonymous
yea but i use wolfram alpha normaly u mind using it sending the link plz
ParthKohli
  • ParthKohli
Because you really are looking to minimise arc-length given the above constraints.
anonymous
  • anonymous
ok wait do i do the same steps as i did for my previous function find a,b,c,d
anonymous
  • anonymous
and then put them in an expression and use arc length to get length
ParthKohli
  • ParthKohli
I think Wolfram is gonna have some good fun with this one.
anonymous
  • anonymous
heheh
ParthKohli
  • ParthKohli
Here, you have a, b, c, d and e.
anonymous
  • anonymous
yea kinda forgot about e
anonymous
  • anonymous
thank you so much for ur help
ParthKohli
  • ParthKohli
I'm curious to see how you'll input so much into Wolfram. It's scary.
anonymous
  • anonymous
what if i find an expression then too
ParthKohli
  • ParthKohli
sorry, the last equation is\[4a(10^3) + 3b(10^2 ) + 2c(10) + d = \color{red}1\]
anonymous
  • anonymous
ok
ParthKohli
  • ParthKohli
wow the expression is way too long... you'll have to be subscribe to pro
anonymous
  • anonymous
i have subscribe pro
ParthKohli
  • ParthKohli
wow, cool.
baru
  • baru
@ParthKohli for arguments sake, (specifics not necessary) i would write the function for arc length squared lets call it L=f(a b c d e) now what?to minimize, we need to substitute variables in terms of each other so that they represent their inter-dependencies right? how will you do that? you have four other equations ? or am i way off, and talking crap?
ParthKohli
  • ParthKohli
yes - we have four equations. in the arc length function, we'll eliminate four variables and will be left with a function that is in terms of only one variable.
baru
  • baru
is that a general thing, as in if there are n unknowns and (n-1) equations we can bring it down to 1 variable?
ParthKohli
  • ParthKohli
I suppose. Let's take a smaller example.\[f(x,y,z) = x+y-z\]\[x=y+1\]\[z = -x\]\[f(x,y,z) =x+(x-1) - (-x) = 3x - 1\]
ParthKohli
  • ParthKohli
Three variables, two equations. Now we can minimise the function with one-variable very easily.
baru
  • baru
ok. thanks :)
baru
  • baru
@ParthKohli one other thing, i get why you have picked an equation with 5 unknowns since we have 5 pieces of information, but the equation could just as easily been \[ax^7 + bx^6 +cx^5 +dx +e\] or something else equally arbitrary so why specifically have you chosen the one that you have?
ParthKohli
  • ParthKohli
that is a very good observation, one that I did give some thought to but skipped. unfortunately, it may be impossible to find *the* polynomial that satisfies the above constraints because the degree may go as high as... anything we want it to reach. the degree-seven polynomial you wrote may not be the best degree-seven polynomial to satisfy the conditions because it's missing some terms and unless we're sure while solving for the seventh-degree that those terms turn out to be zero, we'll never be sure. so choose a degree-four polynomial adds certainty to what we get - we can be sure that we'll get the best fourth-degree polynomial. of course there are polynomials much better than that one, but we can only go as far as we can look.
ParthKohli
  • ParthKohli
ultimately, we're looking for a polynomial that resembles a straight-line the most in the region x = 10 to x = 30, so we can add as many terms as we like, but we can't go that far - can we?
baru
  • baru
ah..makes sense. thanks a lot!!
ParthKohli
  • ParthKohli
by the way, we still can approach this problem through multivariable calculus for polynomials that have more terms than five. that way, we can get an even closer polynomial.
anonymous
  • anonymous
hey its not making sense
ParthKohli
  • ParthKohli
thanks for the question - I really had to think about it to come up with an answer.
anonymous
  • anonymous
5 variables how do i get values for 5 unknown variable
ParthKohli
  • ParthKohli
plug it into WolframAlpha
anonymous
  • anonymous
i only have 4 equation
anonymous
  • anonymous
i found a ,b, c ,d but not e
ParthKohli
  • ParthKohli
were you able to find a, b, c, d in terms of e?
anonymous
  • anonymous
\[0.0001852x ^{4}-0.01246x ^{3}+0.2076x ^{2}-0.01555x+e\]
anonymous
  • anonymous
http://math.bd.psu.edu/~jpp4/finitemath/4x4solver.html
ParthKohli
  • ParthKohli
wait what? you were able to uniquely determine a, b, c, d?!
anonymous
  • anonymous
from here
ParthKohli
  • ParthKohli
it's not a 4x4 equation so
anonymous
  • anonymous
yea i thought so help plz
ParthKohli
  • ParthKohli
plug the stuff into Wolfram and it should give you all other variables in terms of one variable
baru
  • baru
@ayeshaafzal221 where did you find this question?
anonymous
  • anonymous
it didnt
anonymous
  • anonymous
its my practise question for exams
ParthKohli
  • ParthKohli
what did it return?
ParthKohli
  • ParthKohli
you don't have a Wolfram pro subscription do you?
anonymous
  • anonymous
on my ipad yes
anonymous
  • anonymous
even on my ipad not working
ParthKohli
  • ParthKohli
there's this data entry or something feature which you can use
anonymous
  • anonymous
ok
anonymous
  • anonymous
or u know the values i have found cant i just sub it into any equation and get the e value
ParthKohli
  • ParthKohli
no, you can't do that. it's not a 4x4 equation so you most likely input the wrong equations.
anonymous
  • anonymous
yea true i am gonna go old fashion , and do em manually
baru
  • baru
@ParthKohli , when you say multi-variable calculus approach: are you talking about Lagrange multipliers?(i've just reached this topic)
ParthKohli
  • ParthKohli
Obviously. Yes.
baru
  • baru
alright..thanks
anonymous
  • anonymous
@ParthKohli srry i am disturbing u again when u say we have five variables and one of them was minimum arc length what do u mean?

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