SOLUTION: two cities are located in the xy plane at the points A (7,5) and B (13,8). a company wishes to place a factory on a road between the cities at A and B. it is determined thatvthe co

Algebra ->  Human-and-algebraic-language -> SOLUTION: two cities are located in the xy plane at the points A (7,5) and B (13,8). a company wishes to place a factory on a road between the cities at A and B. it is determined thatvthe co      Log On


   

Question 1058769: two cities are located in the xy plane at the points A (7,5) and B (13,8). a company wishes to place a factory on a road between the cities at A and B. it is determined thatvthe company can make the most money if it places the factory exactly beyween A and B. however due to environmental hazards produced by the factory, it is safe for the people living in city A and city B only the factory is located exactly one third of the distance from city A going toward city B. find the coordinates of the point where the factory should be placed to ensure the most ethical outcome.
Found 3 solutions by josgarithmetic, MathTherapy, ikleyn:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Line AB is equation y-5=%28%288-5%29%2F%2813-7%29%29%28x-7%29
y=5%2B%281%2F2%29%28x-7%29
y=x%2F2-7%2F2%2B10%2F2
y=x%2F2%2B3%2F2


Distance from A to B is sqrt%28%2813-7%29%5E2%2B%288-5%29%5E2%29
sqrt%2836%2B9%29
sqrt%2845%29
3sqrt%285%29

You want to know the point on the line so that
%281%2F3%29%283sqrt%285%29%29=sqrt%28%28x-7%29%5E2%2B%28y-5%29%5E2%29
and you want to substitute for y and then solve this equation for x.

sqrt%285%29=sqrt%28%28x-7%29%5E2%2B%28x%2F2%2B3%2F2-5%29%5E2%29

sqrt%285%29=sqrt%28%28x-7%29%5E2%2B%28x%2F2%2B3%2F2-10%2F2%29%5E2%29

sqrt%285%29=sqrt%28%28x-7%29%5E2%2B%28x%2F2-7%2F2%29%5E2%29
square both sides.
5=%28x-7%29%5E2%2B%28x%2F2-7%2F2%29%5E2
5=x%5E2-14x%2B49%2B%281%2F4%29%28x-7%29%5E2
20=4x%5E2-56x%2B196%2Bx%5E2-14x%2B49
20=5x%5E2-70x%2B245
5x%5E2-70x%2B220=0
highlight_green%28x%5E2-14x%2B44=0%29-------not factorable so use general solution for quadratic formula.

x=%2814%2B-+sqrt%2814%5E2-4%2A44%29%29%2F2

x=%2814%2B-+sqrt%28108%29%29%2F2

x=%2814%2B-+sqrt%289%2A4%2A3%29%29%2F2

x=%2814%2B-+12%2Asqrt%283%29%29%2F2

highlight%28x=7%2B-+6%2Asqrt%283%29%29-------Not finished. One of these will work and the other will not work. Most likely the PLUS form will be the correct choice; and then find the corresponding y value from the equation of the line AB.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
two cities are located in the xy plane at the points A (7,5) and B (13,8). a company wishes to place a factory on a road between the cities at A and B. it is determined thatvthe company can make the most money if it places the factory exactly beyween A and B. however due to environmental hazards produced by the factory, it is safe for the people living in city A and city B only the factory is located exactly one third of the distance from city A going toward city B. find the coordinates of the point where the factory should be placed to ensure the most ethical outcome.
This exact problem was submitted a while back. I gave you an answer for it then. Actually, itw's problem # 1042721.

Again, the CORRECT answer is: the coordinate point that is 

I hope you know by now to IGNORE the other person's solution, if it can be called that. I hope you avoided being EXTREMELY CONFUSED with that person's so-called attempt.

If you follow that person, you're going to be so lost that not even the National Guard will be able to find you. lol

It's just a simple, simple solution that involves 2 SIMILAR TRIANGLES. That's all. Try it!! You can do it!

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
The solution by the tutor "MathTherapy" is correct.