Someone good at math triangles and circles?

[Euphoriet]

Sorry to bother you guys.. but if there is someone out there that is good at math and wouldn't mind helping me.. could you aim me sn:"euphoriet" thanks.

 

[Tide]

It seems you could get substantial help right here.

 

[Euphoriet]

I would rather talk on aim for this one... and mainly... i want to make a friend that is good at math ;) =-P

But here it is:

Starting at the same spot on a circular track that is 80 meters in diameter, Sandy and Candy
run in opposite directions, at 300 meters per minute and 240 meters per minute, respectively.
They run for 50 minutes. What distance separates Sandy and Candy when they finish? There is
more than one way to interpret the word distance in this question.

That last part has me... i know there are many ways.. but I have most trouble with the straight line distance one.

 

[Tide]

Divide each runner's total distance traveled by the radius of the track. This gives the angle (in radians) about the center of the circular track that each has traversed. Account for the fact they ran in opposite directions by making one positive and the other negative (i.e. one ran counterclockwise and the other clockwise).

If R is the radius of the track and they started, say, at x = R and y = 0 then Sandy's position would be $(R \cos \theta _s, R \sin \theta _s)$ and Candy's would be $(R \cos \theta _C, R \sin \theta _C)$.

To find the straight line distance find the difference between their x coordinates and the difference between their y coordinates (call them $\Delta x$ and $\Delta y$, for example. Their separation is then simply
$$d = \sqrt {{\Delta x}^2 + {\Delta y}^2}$$

 

[Euphoriet]

So far i have 59.52 for Sandy.. and 47.75 for Cyndy.. their distance in relation to circumference is like 11,77 right?... but .. i really don't know how to solve for their distance in a straight line.. and I am sure I got all this wrong... so yeah...

 

[Euphoriet]

wait.. its actually .23 of the total track... 80xpi x 0.23 gives the circular distance... but I need the straight line still

 

[TenaliRaman]

Did u follow Tide's approach , he almost has the entire answer for u there...

Go along the steps Tide suggested and if any problem, post ur working herer so u can be better helped...

-- AI

 

[.....]

i get 25.46m...could be wrong...this is how i did it though:

first you need to work out how where they are when they stop running

for sandy, 300 meters per minute for 50 minutes gives 150,000m as the total distance she's run

divide by 80 to get the number of laps she's run - you get 187.5 - so she's halfway around the track from where she started

now do the same for candy and you get 150 laps exactly - so she ends up back where she started from.

If you're following the circular path, then sandy and candy must be 40m away from each other.

but, i assume the "more than one way to interpret distance" thing means it wants the distance between the two as a straight line.

this distance will be the diameter since one is exactly halfway around the track from the other.

you know 80m is the circumference, and you know that the circumference of a circle is 2 X Pie X Raduis (call pie P and radius R for ease's sake)

80 = 2PR
R = 80/(2P)
= 12.73

diameter is 2 times R

D = 2 X 12.73
= 25.46

 

[HallsofIvy]

I would interpret "What distance separates Sandy and Candy when they finish? There is more than one way to interpret the word distance in this question." as meaning either that would accept either circular or straight distance or that they want both answers. I certainly wouldn't interpret "There is more than one way to interpret the word distance" as meaning they were insisting on specific one.

 

[.....]

I could have sworn i wrote "aswell"

oh well...lol...

 

[Euphoriet]

I know he gave me the path to the answer but I don't know what the s and the c in front of the cos and sin mean... I would solve it myself but I am having trouble wih that.. also what would I replace for the theta symbol... I have no other experience with this "variable"

 

[Tide]

S = Sandy
C = Candy