Question

I am doing a paper where I am trying to find the optimum spot in the world to put a power generator. To do this, I am taking the latitude longitude coordinates of different cities and finding the optimum place for the generator. I planned to use the coordinates (x,y) to relate to the coordinates of the cities, then using the distance formula between (x,y) and each of the ten cities. I was then going to add each distance together for a total distance. I want to minimize this. My first thought was to take the first derivative and set it equal to zero, but I can't with all my variables.

Answer 1

so you are saying you have 10 cities, and you want to put the generator in the middle of them to minimize the power lost in the transmission lines? do I got this right?

Answer 2

You are correct

Answer 3

alright,
and one last question, are you considering how much power each city will need, or are you considering them all needing the same amount of power (or inother words, the population for each of 10 cities is identical)?

Answer 4

No...I might consider that later, but for right now, I just want to consider location. 2D. I might also do 3D later as well.

Answer 5

Hello Everybody, I am in 3rd grade and for science there is a question asking me how parents make a baby? I dont know plz help. i will give a medal

Answer 6

ok, so we will ignore how much each population needs..

to find the optimum location being an x,y location, we will need to take all the x values from each city, add them together then divide by 10.

then we will take all the y values from each cities coordiante, add them up, and divide by 10.

the value for x and for y will be the 'optimum' location for the generator.

does this make sense?

Answer 7

so lets say I had only 3 cities:

(5, 9)
(12, 13)
(11, 2)

(5+12+11) / 3 = 9.33
(9+13+2) / 3 = 8
so, the location of my generator would be (9.33, 8)

Answer 8

Yes, so I don't need calculus? Haha... that would have been nice to know. Thank you so much! Can I translate this into 3D geometry, using the world as a sphere and finding the optimum location from there?

Answer 9

And what if I did factor in population of the cities?

Answer 10

"Can I translate this into 3D geometry, using the world as a sphere and finding the optimum location from there?"

ya, just use the elevation above sea level for each city.
so
(5, 9, 1220)
(12, 13, 2300)
(11, 2, 2200)

(1220 + 2300 + 2200) / 3 = 1906.67

so, the coordinate would be
(9.34, 8, 1906.67)

but this idea only works if we consider a plane surface, if you wanted to use a sphere, then the generator would end up being in the middle of the sphere, i think you can figure out why

Answer 11

Thanks! Except, what if I just considered surface of the sphere and not the inside? Because straight lines are much different on a sphere than on a 2D surface.

Answer 12

"And what if I did factor in population of the cities?

so lets have the third number be the population of people
(5, 9, 500000) so, 500,000 people live at city (5,9)
(12, 13, 250000)
(11, 2, 112000)

then if I wanted to find the optimum x, y locaiton of the generator, it would go like this:

\[\frac { 5(500000) + 12(250000) + 11(112000) } {500000+250000+112000} = 7.8\]

make sense?

Answer 13

Yes, this makes sense. Are there any other angles that you think I could explore this topic that involve calculus? It's just that I want to try to make this more intricate than just averages.

Answer 14

and for y, it would be:

\frac {9(500000)+13(250000)+2(112000)} {500000 + 250000 + 112000} = 9.25

so, the location, considering population would be:
(7.8, 9.25)

Answer 15

My gosh! Thank you so much! I appreciate everything you are doing

Answer 16

"Yes, this makes sense. Are there any other angles that you think I could explore this topic that involve calculus? It's just that I want to try to make this more intricate than just averages."

hmm...
calculus focuses on 'rate of change' so, if you wanted to include calculus into this, you would need to think of either population changes with time, or how power changes with time.

Answer 17

Well, my plan was to take the derivative of the distance formulas added together and set it equal to zero, finding a global minimum of the function. That was unnecessary as I see now

Answer 18

Wait! What if I did all that you said to do, but did it from the angle of the largest cities in 1900, 1910, 1920....up to 2010? Then I could possible develop a model...maybe...

Answer 19

I see,

ya, I think that could work.
or maybe you could take the those years, make a model, then project the model into the future to 2060 or something

Answer 20

Yes! Whoever you are, you just really helped me. Thank you!

Answer 21

haha
ur welcome

Answer 22

well message me if you got anymore questions

have fun with your model.

Answer 23

Thanks