Shortest path possible

[Polonious]

Hi all,
I'm stuck with this problem i cannot solve.

Let a point O be the origin of xy-coordinate plane. We define four points A(1, 0), B(1, 1), C(2, 1), D(3, 1) on the plane. Let us start from C, go through a point on line OA, go through a point on OB, and reach D with the mínimum lenght of the path. In this path, the pint on the line OA is [(Answer 1), (Answer 2)], that on the line OB is [(Answer 3), (Answer 4)] and the lenght of the path is [(Answer 5)].

I tried finding the formula of the distance and then derive it to find the minimum, but I get stuck with two variables. Any ideas?

 

[Otis]

Hi Polonius. Have you plotted the given points and thought about a path? Where they say, "go through a point", I'm thinking that just touching a point (without crossing a line) is allowed. Please share your work so far. Thanks!



[imath]\;[/imath]

 

[The Highlander]

Hi all,
I'm stuck with this problem i cannot solve.

Let a point O be the origin of xy-coordinate plane. We define four points A(1, 0), B(1, 1), C(2, 1), D(3, 1) on the plane. Let us start from C, go through a point on line OA, go through a point on OB, and reach D with the mínimum lenght of the path. In this path, the pint on the line OA is [(Answer 1), (Answer 2)], that on the line OB is [(Answer 3), (Answer 4)] and the lenght of the path is [(Answer 5)].

I tried finding the formula of the distance and then derive it to find the minimum, but I get stuck with two variables. Any ideas?

Hi Polonius,

When you say you tried to "derive it", I presume you meant differentiate it but I'm not sure any Calculus is required to solve this problem.

As Otis says, your starting point must be to draw a graph showing the x & y axes and plotting the points A, B, C & D.
Then starting from C you have to go down to the x-axis, up to the (extended) line OB and then back down again to the point D.

A phase shifted & elevated sine curve (∿) might do that but would that give you the shortest path?
(Your question doesn't say the path has to be part of a continuous curve. )

 

[Dr.Peterson]

(Your question doesn't say the path has to be part of a continuous curve. )

I think "path" does imply continuous, but not smooth! In fact, I expect it to be piecewise linear and connected.

I tried finding the formula of the distance and then derive it to find the minimum, but I get stuck with two variables. Any ideas?

You put this under geometry, but apparently expect to use calculus. There may be a purely geometrical approach, but you presumably mean writing a formula for the total length of three segments in terms of two variables locating the points on OA and OB, and using differentiation to minimize the sum. Am I right?

We really need to see your actual work, not a general description, in order to help. What formula did you use? What calculus have you learned? I would expect to use multivariable calculus, but there is probably a way to do without special knowledge of that.

Anything you can tell us about the context of the problem (e.g., if it is from a textbook, what topic was just taught?) so we can better guess what you are expected to do.

 

[Dr.Peterson]

I can add that there is in fact a very easy geometrical solution. If you want that rather than calculus, the key word is "reflection".

 

[Polonious]

Hi all, thanks for all your replies. Here's a picture of what I was able to do so far (this is a clean copy i have tried a lot of approaches without success).

And yes, I wanted to "differentiate". English is not my first language, sorry for the mistakes.

What I want to do is to find a relation between variable "k" and "j" to reach a formula of the distances with just one variable and apply differentiation to find the value of either "k" or "j" that minimizes the distance function.

But, reading your comments, I may be trying to solve this the wrong way. I will search about reflection and see if it helps me.

This is a pre-university exercise, so I presume there is no need of multiple-variable differentiation.

 

[The Highlander]

Hi all, thanks for all your replies. Here's a picture of what I was able to do so far (this is a clean copy i have tried a lot of approaches without success).

And yes, I wanted to "differentiate". English is not my first language, sorry for the mistakes.

What I want to do is to find a relation between variable "k" and "j" to reach a formula of the distances with just one variable and apply differentiation to find the value of either "k" or "j" that minimizes the distance function.

But, reading your comments, I may be trying to solve this the wrong way. I will search about reflection and see if it helps me.

This is a pre-university exercise, so I presume there is no need of multiple-variable differentiation.

Hi Polonius,

My first thought was to extend the line OB (ie: y=x), just as you have done, but it was suggested to me that this may not be "allowed". If that's true then a similar "restriction" might apply to OA too.

How might that affect your consideration of the path?

 

[Dr.Peterson]

Hi all, thanks for all your replies. Here's a picture of what I was able to do so far (this is a clean copy i have tried a lot of approaches without success).

And yes, I wanted to "differentiate". English is not my first language, sorry for the mistakes.

What I want to do is to find a relation between variable "k" and "j" to reach a formula of the distances with just one variable and apply differentiation to find the value of either "k" or "j" that minimizes the distance function.

But, reading your comments, I may be trying to solve this the wrong way. I will search about reflection and see if it helps me.

This is a pre-university exercise, so I presume there is no need of multiple-variable differentiation.

Thanks.



This is probably what I would do if I wanted to use (single-variable) calculus. You should be able to minimize in two steps, perhaps first finding j to minimize the total distance for any fixed value of k (which would find j in terms of k), and then finding k to minimize that. But it doesn't look pretty!

If this was given in a calculus context, then go ahead and try that; but if it was in a broader context, and if geometry (or perhaps optics?) was part of the curriculum, then you might think about reflection. I'm not sure where that idea might be taught; I think of it as a useful trick just for this sort of problem. Here is a reference to the basic idea: https://www.maa.org/book/export/html/428409

 

[Polonious]

I suppose it should be allowed because I have the answers and the line "bounds" on points similar to the ones I draw.

By the way, k=7/4, j=7/5 and the distance is sqrt(17). The thing is I don't know how to reach those values.

 

[Polonious]

Thanks.

View attachment 31420

This is probably what I would do if I wanted to use (single-variable) calculus. You should be able to minimize in two steps, perhaps first finding j to minimize the total distance for any fixed value of k (which would find j in terms of k), and then finding k to minimize that. But it doesn't look pretty!

If this was given in a calculus context, then go ahead and try that; but if it was in a broader context, and if geometry (or perhaps optics?) was part of the curriculum, then you might think about reflection. I'm not sure where that idea might be taught; I think of it as a useful trick just for this sort of problem. Here is a reference to the basic idea: https://www.maa.org/book/export/html/428409

Thanks for the link. I will investigate it.

The problem does not restrict the solution to one area of maths, so any plausible solution is welcome.

This is part of a past exam taken to get a MEXT undergraduate scholarship. MEXT being the Ministry of Education (among other things) of Japan.

The thing is they don't give an official list of topics you have to study to take it. You have to look at past exams and derive the topics from them. That's why i didn't even know where to post this thread, because I didn't even know which topic this belongs to.

By the way, I'm an engineer and althoug it's been some years since I last used some maths I, for sure, never studied about reflection...

 

[Polonious]

I was able to solve it with the reflection method. It was, in fact, quite simple and elegant.

Thank you all for giving me the right tools to solve the problem.

 

[Dr.Peterson]

I was able to solve it with the reflection method. It was, in fact, quite simple and elegant.

Thank you all for giving me the right tools to solve the problem.

Excellent. Just for fun, here is a drawing of my reflection solution (omitting the math to get the actual numbers, which agree with yours):

​

 

[Polonious]

I leave here my solution. I reflected it in another way (just reflected D, two times). I see that your way is even mor direct.