MPM 2D - Lead up Task #1 - 2008-2009
Pipeline Relay Station
The Stench Gas Company wishes to run gas lines to two towns, Sulphurville and Beanville, from a major pipeline that runs near both towns as shown in the diagram. The Gas Company can afford to build only one relay station on the major pipeline. Therefore, it wants to locate the relay station at a point where the two gas lines total length to the towns would be a minimum. Your job is to determine the location of the relay station.
(Note: The diagram is not to scale and for illustration only. The proper location of the relay station still needs to be determined.)
The technique to find the relay station point (that is the minimum distance to the two towns) is as follows:
1. Reflect the point that represents Beanville through the line representing the pipeline
(i.e. the pipeline is the perpendicular bisector of the segment connecting Beanville and its reflection point).
2. Connect this reflection point with Sulphurville. This line segment intersects the pipeline at the needed relay station point.
Part A – Group work
i) What information do you need in order to find the location of the relay station?
ii) Discuss the steps you are going to take to solve this problem. Be sure to explain your thinking, show your work and include all formulae required.
Part B – Individual work
The diagram has been placed on a grid. The equation of the major gas pipeline is . Sulphurville is located at coordinates and Beanville is located at coordinates .
Determine the exact location of the relay station, clearly explaining all of the steps. Be sure to explain your thinking, justify your choices, and include all formulae required.
Note: If you need to round throughout the problem, keep a minimum of two decimal places. Round your final answer to the nearest one decimal place.
MPM 2D - Lead up Task #1 - 2008-2009
Pipeline Relay Station - SOLUTION
B - Beanville
S - Sulferville
P - Pipeline
B' - Reflected point of B across pipeline
M - Midpoint of line BB'
PART B
Step 1: Determine the equation of line BB' in the form
Equation of line BB' is
Step 2: Determine the point of intersection M between line BB' and line P
The point of intersection M between line BB' and line P is: (1.6, -0.2)
Step 3: Given the endpoint B and its midpoint M, determine the coordinates of the other endpoint B' using the midpoint formula
The coordinates of point B' are: (3, -3)
Step 4: Determine the equation of line B'S
The equation of line B'S is:
Step 5: Determine the point of intersection between lines B'S and P using substitution.
The point of intersection between lines B'S and P is (4, 1).
Therefore, the relay station should be at (4,1)
Original task courtesy of Ron Gaudreau