Help! How to Solve Math Methods Problem Involving Radio Transmitter & Railway

[Astronomer107]

I'm in IB math methods, working on a project and I can't seem to figure out what to do with this problem.

A radio transmitter sends signals to a railway which run along a straight track. When a set of coodinate axes is used to represent this system, the transmitter is at R(1,0), and the track T is a line w/ equation 2x+y=30, where the units are kilometers.
- Points A and be are the points of T that cross the x and y axis
- The engine receives the strongest signal at C, which is the point on T closest to R.
- P(x,y) is a general point on T

My problem is this:
let s(x) represent the distance between R(1,0) and P(x,y). Show that
s(x)= sq rt (5x^2-122x+901).

Help! I don't even know where to begin. I said that point C is
(15x-15, -60x + 900). Since RP is (x-1)i + (-2x+30)j, does the 901 in the thing I'm supposed to show have to do with... oh never mind I'm REALLY confused. Please help me. Thanks!

 

[mrbill]

poof

First, all we have to do is show that s(x) = sqrt (5x^2 - 122x + 901). It seems like your making this out to be a lot more complicated then it really is. Forget vectors or anything other then good old algebra for a minute.

Recall the distance formula: sqrt[ (x2-x1)^2 + (y2-y1)^2 ]

We plug in our values for (x1,y1) (x2,y2) from the points given.

Therefore:

s(x) = sqrt [ (x-1)^2 + y^2 ]

Now we need to change that y^2 into x..we are in luck because they have given us the equation of the line on which these points are located. Solve 2x + y = 30 for y... we get: y = 30 - 2x

Substitute this back into the equation we just obtained with the distance formula and multiply it out.

s(x) = sqrt [ (x-1)^2 + (30-2x)^2 ]

s(x) = sqrt [ x^2 - 2x +1 + 900 - 120x + 4(x)^2 ]

simplify:

s(x) = sqrt [ 5(x)^2 - 122x + 901 ]

There you have it.

 

[Astronomer107]

Thanks!

Thank you so much! I can't believe how simple that was. I have such a huge problem when I am asked to prove things. I just never seem to find the way to do it.

 

[Astronomer107]

Another Problem

In the above problem, they want me to draw an arc with the center R with a radius of 28 and then find the length of the portion of T that will be within the range of the arc. The drawing the arc was no problem, but I don't know how to find it.