Question

an island has three towns located at points with the coordinates a (15,3) b (25,10) and c (0,20) where the coordinates have units of Km the government wants to locate a toxic waste dump at a position as far as possible away from each town you may assume that this location is not on the edge of the island find the coordinates at the point where the toxic waste dump should be located

Answer 1

9514 1404 393

Answer:

  (13.06, 16.41)

Step-by-step explanation:

Any point not on a perpendicular bisector of the line between towns will be closer to one town than the other. So, the intersection of perpendicular bisectors will be the equidistant from all towns. That point is the circumcenter of the circle circumscribing the triangle with the towns as its vertices. This can be found as the intersection of two perpendicular bisectors of the segments between the towns.

The midpoint of AB is ...

  (A +B)/2 = ((15, 3) +(25, 10))/2 = (40, 13)/2 = (20, 6.5)

The midpoint of AC is ...

  (A +C)/2 = ((15, 3) +(0, 20))/2 = (15, 23)/2 = (7.5, 11.5)

__

The differences between points are ...

  B -A = (25, 10) -(15, 3) = (10, 7)

  C -A = (0, 20) -(15, 3) = (-15, 17)

Then the perpendicular bisector equations can be written as ...

  bisector of AB  ⇒  10(x -20) +7(y -6.5) = 0

     10x +7x -245.5 = 0

  bisector of AC  ⇒  -15(x -7.5) +17(y -11.5) = 0

     -15x +17y -83 = 0

These have coefficients that aren't particularly nice, so a method similar to Cramer's Rule is suitable for solving this pair of equations. Using the "cross multiplication method", we have ...

  x = (7(-83) -17(-245.5))/(10(17) -(-15)(7)) = 3592.5/275 = 13 7/110 ≈ 13.06

  y = (-245.5(-15) -(-83)(10))/275 = 4512.5/275 = 16 9/22 ≈ 16.41

The point as far as possible from each town within the boundary of the island has coordinates (13.06, 16.41).

_____

Further detail regarding the solution

Here, we have written the equation for the perpendicular bisector this way. The difference in coordinates between points R and S can be called ...

  R -S = (∆x, ∆y)

The midpoint of segment RS can be called ...

  (R +S)/2 = (h, k)

Then the line that is a perpendicular bisector of RS can be written as ...

  ∆x(x -h) +∆y(y -k) = 0

This is the form that is used above.

__

The "cross multiplication method" for solving a pair of general-form linear equations ...

  ax +by +c = 0

  dx +ey +g = 0

will give the solution as ...

  1/(ae -db) = x/(bg -ec) = y/(cd -ga)

If you write the coefficients in two rows of four, the pattern of products and differences is easy to see.

  $\left[\begin{array}{cccc}a&b&c&a\\d&e&g&d\end{array}\right]$