Function describing distances in term of time.

A vehicle driving east at 60 mph and another vehicle B drives north at 35 mph crossed each other's path. From where their paths crossed, write the function that tells the distance that they are from each other in terms of time elapsed.

Well from what I understand is that they intersect, I believe. So one travels 60 mph east, while the other drives 35 mph north.

Though, I'm not really sure how to start off.

I am giving the speed, which I believe is the slope?

And then they perpendicularly intersect right?

I just got back to school and kind of forgot how to do some of the steps and wanted to review.

So I would need to find the equation of where they intersected right? With the variables of slope, and something else. I kind of need help getting started off.

Re: Function describing distances in term of time.

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Originally Posted by

**Chaim** A vehicle driving east at 60 mph and another vehicle B drives north at 35 mph crossed each other's path. From where their paths crossed, write the function that tells the distance that they are from each other in terms of time elapsed.

Well from what I understand is that they intersect, I believe. So one travels 60 mph east, while the other drives 35 mph north.

Though, I'm not really sure how to start off.

I am giving the speed, which I believe is the slope?

And then they perpendicularly intersect right?

I just got back to school and kind of forgot how to do some of the steps and wanted to review.

So I would need to find the equation of where they intersected right? With the variables of slope, and something else. I kind of need help getting started off.

**Hint**

Let us call the vehicle moving east at 60mph $\displaystyle A$ and the vehicle moving north at 35mph $\displaystyle B$ who have position vectors $\displaystyle \textbf{A}$ and $\displaystyle \textbf{B}$ respectively. Clearly from the information, we can deduce that the position vectors $\displaystyle \textbf{A}, \ \textbf{B}$ of the vehicles $\displaystyle A, \ B$ after $\displaystyle t$ hours will be given by:

$\displaystyle \textbf{A} = 60t \ \textbf{i} + 0t \ \textbf{j}$

$\displaystyle \textbf{B} = 0t \ \textbf{i} + 35t \ \textbf{j}$

Now Pythagoras.

Re: Function describing distances in term of time.

Quote:

Originally Posted by

**FelixFelicis28** **Hint**

Let us call the vehicle moving east at 60mph $\displaystyle A$ and the vehicle moving north at 35mph $\displaystyle B$ who have position vectors $\displaystyle \textbf{A}$ and $\displaystyle \textbf{B}$ respectively. Clearly from the information, we can deduce that the position vectors $\displaystyle \textbf{A}, \ \textbf{B}$ of the vehicles $\displaystyle A, \ B$ after $\displaystyle t$ hours will be given by:

$\displaystyle \textbf{A} = 60t \ \textbf{i} + 0t \ \textbf{j}$

$\displaystyle \textbf{B} = 0t \ \textbf{i} + 35t \ \textbf{j}$

Now Pythagoras.

I see, thanks.

Though I'm a bit confuse. What I'm imagining on my head is a perpendicular line.

I'm not sure if I'm positively correct, but what I do with those two equations is make the triangle, then find the point where they insect which is at the bottom right? Like this:

-/ |

/_| <--- This part? (Sorry, ugly interpretation of a triangle)

Re: Function describing distances in term of time.

Quote:

Originally Posted by

**Chaim** I see, thanks.

Though I'm a bit confuse. What I'm imagining on my head is a perpendicular line.

I'm not sure if I'm positively correct, but what I do with those two equations is make the triangle, then find the point where they insect which is at the bottom right? Like this:

-/ |

/_| <--- This part? (Sorry, ugly interpretation of a triangle)

Yes - just to clarify, the question assumes that the two vehicles have *already* intersected at a right angle and the question asks for the distance between the vehicles at time $\displaystyle t$ after they've intersected, right?

So it'll look a bit more like this:

\

.\

..\

...\

....\

___.\

but you get the point. :P

Re: Function describing distances in term of time.

Quote:

Originally Posted by

**FelixFelicis28** Yes - just to clarify, the question assumes that the two vehicles have *already* intersected at a right angle and the question asks for the distance between the vehicles at time $\displaystyle t$ after they've intersected, right?

So it'll look a bit more like this:

\

.\

..\

...\

....\

___.\

but you get the point. :P

Ah, I see.

So lets see.

Well it asks to write a function about the distance they are from each other in terms of time.

So lets see if I get this.

Pythagoras' equation is a^{2}+b^{2}=c^{2}

So that means 60 would be a, and 35 would be b?

Therefore c would equal around 69.4622.

That means it would be y=69.4622 + b, and then I would find b for that?

Re: Function describing distances in term of time.

Quote:

Originally Posted by

**Chaim** Ah, I see.

So lets see.

Well it asks to write a function about the distance they are from each other in terms of time.

So lets see if I get this.

Pythagoras' equation is a^{2}+b^{2}=c^{2}

So that means 60 would be a, and 35 would be b?

Therefore c would equal around 69.4622.

That means it would be y=69.4622 + b, and then I would find b for that?

Not quite. Do you know how to find the distance between two vectors?

The question wants a function for the distance between the vehicles in terms of time, hence why we parameterize the positions of $\displaystyle A$ and $\displaystyle B$ in terms of time, $\displaystyle t$.

Re: Function describing distances in term of time.

Quote:

Originally Posted by

**Chaim** Ah, I see.

So lets see.

Well it asks to write a function about the distance they are from each other in terms of time.

So lets see if I get this.

Pythagoras' equation is a^{2}+b^{2}=c^{2}

So that means 60 would be a, and 35 would be b?

Therefore c would equal around 69.4622.

That means it would be y=69.4622 + b, and then I would find b for that?

I'm heading off now so I'll leave you with this.

Tbh, I may have confused you by introducing vectors, sorry. :P The point is, we want a function of the position of the vehicles in terms of time, hence we parameterize the position of the vehicles.

If A is travelling east at 60mph then clearly after 1 hour, A has travelled 60 miles east, after 2 hours 120 miles east, 3 hours 180 miles east $\displaystyle \cdots$ after $\displaystyle t$ hours, $\displaystyle 60t$ miles east and $\displaystyle 0t$ miles north.

Similarly for B, after $\displaystyle t$ hours, it has travelled $\displaystyle 35t$ miles north and $\displaystyle 0t$ miles east.

Now pythagoras but **keep** your parameter (i.e. $\displaystyle t$) when you use pythagoras and you're done.

Re: Function describing distances in term of time.

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Originally Posted by

**FelixFelicis28** I'm heading off now so I'll leave you with this.

Tbh, I may have confused you by introducing vectors, sorry. :P The point is, we want a function of the position of the vehicles in terms of time, hence we parameterize the position of the vehicles.

If A is travelling east at 60mph then clearly after 1 hour, A has travelled 60 miles east, after 2 hours 120 miles east, 3 hours 180 miles east $\displaystyle \cdots$ after $\displaystyle t$ hours, $\displaystyle 60t$ miles east and $\displaystyle 0t$ miles north.

Similarly for B, after $\displaystyle t$ hours, it has travelled $\displaystyle 35t$ miles north and $\displaystyle 0t$ miles east.

Now pythagoras but **keep** your parameter (i.e. $\displaystyle t$) when you use pythagoras and you're done.

Oh now I think I got it!

So basically it's to find at which hour they intersect, then use Pythagoras because you have the x and y, right?

Thanks!

That would find c, which would be the distance of the long side.

And then all I would just need to do is put in the variable t somewhere?

Re: Function describing distances in term of time.

Quote:

Originally Posted by

**Chaim** Oh now I think I got it!

So basically it's to find at which hour they intersect, then use Pythagoras because you have the x and y, right?

No. The question assumes they have *already* intersected and is asking you to derive a function for the distance between the vehicles in terms of time *after* they have intersected.

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Thanks!

That would find c, which would be the distance of the long side.

And then all I would just need to do is put in the variable t somewhere?

No - you can't just stick the parameter $\displaystyle t$ in somewhere (you can I suppose in this case but it becomes bad practice imo once you do harder questions). We have started the question by parameterizing the positions of the vehicles with time $\displaystyle t$. When you use pythagoras to work out the distance between the vehicles, *keep* your parameter when you use pythagoras and you will get a formula for the distance between the vehicles in terms of $\displaystyle t$.

Re: Function describing distances in term of time.

Quote:

Originally Posted by

**FelixFelicis28** No. The question assumes they have *already* intersected and is asking you to derive a function for the distance between the vehicles in terms of time *after* they have intersected.

No - you can't just stick the parameter $\displaystyle t$ in somewhere (you can I suppose in this case but it becomes bad practice imo once you do harder questions). We have started the question by parameterizing the positions of the vehicles with time $\displaystyle t$. When you use pythagoras to work out the distance between the vehicles, *keep* your parameter when you use pythagoras and you will get a formula for the distance between the vehicles in terms of $\displaystyle t$.

Oh wait... now I understand now...

It was just sqrt(60^{2}t+35^{2}t)

That was the function... xD

My bad, I overthought it.

So it was d(t) = √(60^{2}+35^{2}t) And that is the complete function.

Thanks for your help :D

Re: Function describing distances in term of time.

Quote:

Originally Posted by

**Chaim**

Oh wait... now I understand now...

It was just sqrt(60^{2}t+35^{2}t)

That was the function... xD

My bad, I overthought it.

So it was d(t) = √(60^{2}+35^{2}t) And that is the complete function.

*Almost*. It should be $\displaystyle \sqrt{(60t)^2 + (35t)^2}$. As I've said - if you parameterize the positions of vehicle A and B, then you need to include the parameter when you use pythagoras i.e. if the $\displaystyle d$ is the distance between the vehicles then $\displaystyle d^2 = (60t)^2 + (35t)^2$.

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Thanks for your help :D

No problem. :)