Habluka's Weekly Challenge #2

[Habluka]

Since there's another thread with relatively simple problems that people are posting in general, I've decided to make the ones in my threads slightly more interesting now, which means it's possible that more people will not be able to solve them. Also, solutions from writing programs will be harder to get now.

Anyway, I would like to congratulate the following people for getting correct answers for one or more of last week's problems: Aekorn, Sarath, Sephr, Lantry (even though I don't like his answer), and also Roflcopter even though he didn't actually give an answer (his post was relevant enough to tell he knew the answer)

Now for this week's problems:

1. • For a given parametric equation (x(t), y(t)), there is a parametric equation (x1(t), y1(t)) where for each value of t, (x1(t), y1(t)) is the end point of a perpendicular line segment to the first equation with a starting point (x(t), y(t)) and a length d. Assuming d(t) is a constant and both equations can be written as functions of y in terms of x, what is the area of the shape you get if you trace the movement of the line segment between (x(t), y(t)) and (x1(t), y1(t))?
     
   • In the case where x(t) = t and y(t) = sin(t), what is the maximum value of d for which an area can be found using this method?
     Here's an image that should help visualize what sort of thing this problem is asking
     Spoiler

     
   
   
2. Let f(x) be a linear function and let Vector V have a magnitude m and a direction theta. Find the orthogonal projection of V onto f(x).
   

[Roflcopter]

Let f(x) be a linear function and let Vector V have a magnitude m and a direction theta. Find the orthogonal projection of V onto f(x).

I'll have a go at them later, but can we assume that we're working in R^2 since you say V has a direction θ? You'd need two angles given the magnitude to define a vector in R^3 (unless the magnitude is 0).

[MykeGregory]

*Maths 'n' shit*

i love lamp.

[DES|Anderson]

Why is everyone doing Habluka's math homework?

[Habluka]

Why is everyone doing Habluka's math homework?

Since I graduate in a couple weeks and the only math class I'm currently taking is Stats, you can safely assume this isn't the case.

[paulmuaddibKA]

Hey, I dont know the answer, but I got two friends who do!

Spoiler

XD

[Bullpup]

Did you mean: For a given parametric equation (x(t), y(t)), there is a parametric equation (x1(y), y1(t))

indeed habluka, a new low point, getting the troglodytes in acf to do your homeworx. Golf clap.


/end(sarc)

[Habluka]

Did you mean: For a given parametric equation (x(t), y(t)), there is a parametric equation (x1(y), y1(t))

indeed habluka, a new low point, getting the troglodytes in acf to do your homeworx. Golf clap.


/end(sarc)

You can have it as x1(t) and y1(t), x1(x) and y1(y), and even x1(x(t)) and y1(y(t)). Answer should still be the same as long as the curve is what it is supposed to be no matter what you write it in terms of.

[Bullpup]

I stand erect.

[Link]

This is beyond my Algebra 1 honors standards.

[Armed]

This is beyond my Geometry honors standards.

[Habluka]

I'm kind of disappointed that no answers have been posted so far for at least the first question after the fast responses from last week, so here's a hint: use trapezoids

I guess these were a little difficult though since most people probably haven't taken calculus or linear algebra though, but at least try to work through them.

[#M|A#Wolf]

The answer is 3 trapezoids and a half.

Here's a picture of a unicorn.

[Snoosnoo]

The answer is 3 trapezoids and a half.

http://www.youtube.com/watch?v=8l73_1oPKoQ

[Roflcopter]

I'm kind of disappointed that no answers have been posted so far for at least the first question after the fast responses from last week, so here's a hint: use trapezoids

I guess these were a little difficult though since most people probably haven't taken calculus or linear algebra though, but at least try to work through them.

I've taken multivariate calculus and linear algebra but the first question is rather ambiuous and the second is just a definition.

For instance you say "there is a parametric equation (x1(t), y1(t)) where for each value of t, (x1(t), y1(t)) is the end point of a perpendicular line segment to the first equation with a starting point (x(t), y(t)) and a length d". I think what you're saying is that we're to take a straight line segment with start (x, y) and end (x1, y1) such that this is perpendicular to the original curve and has length d. But there are two such pairs of functions (x1, y1) that would satisfy this (one "under" and one "above").

I'm also not comfortable that I understand what you mean by trace "the movement of the line segment[s]".

Also see my earlier question where you don't state that we're working in R^2 but seem to assume it by giving us that V has a direction θ.


Anyway here is the argument I think you're looking for:

1. Put .

What you're asking us to compute if I understand correctly (I had to just provide links in a code block or something breaks them):

[SELECT ALL] Code:

http://latex.codecogs.com/png.latex?\lim_{\Delta%20\to%200}%20{\sum_{x%20\in%20I}^{%20}%20\frac{[y%28x+\Delta%29-y%28x%29]d+[y_1%28x+\Delta%29-y_1%28x%29]d}{2}}
http://latex.codecogs.com/png.latex?=%20\frac{d}{2}%20\lim_{\Delta%20\to%200}%20{%20\sum_{x%20\in%20I}^{%20}%20{y%28x+\Delta%29-y%28x%29+y_1%28x+\Delta%29-y_1%28x%29}}
http://latex.codecogs.com/png.latex?=%20\frac{d}{2}%20\lim_{\Delta%20\to%200}%20{%20\sum_{x%20\in%20I}^{%20}%20\frac{y%28x+\Delta%29-y%28x%29+y_1%28x+\Delta%29-y_1%28x%29}{\Delta}%20\Delta}
http://latex.codecogs.com/png.latex?=%20\frac{d}{2}%20\lim_{\Delta%20\to%200}%20{%20\sum_{x%20\in%20I}^{%20}%20[y%27%28x%29+y_1%27%28x%29]%20\Delta}
http://latex.codecogs.com/png.latex?=%20\frac{d}{2}%20%28y+y_1%29

And you can see where this is going...

2. See definition 1.1.

[Alien]

math flames, wth ?? :P where is my favorite apple discussions ? :P

[Roflcopter]

math flames, wth ?? :P where is my favorite apple discussions ? :P

I don't know how you're reading my post to see it as flaming. I'm just not sure whether I have understood what's being asked.

[Bullpup]

are you sure that's not an excerpt from the Necronomicon