Where is the gas station?

Algebra Level 4

There are 11 buildings alongside a straight highway. We need to select only one of building to be a community gas station. To save money and gas, the location of the gas station should be chosen so that the total sum of all distances between each building and the gas station is minimum.

For the following configuration, where is the best building for the gas station?

B C D E F G H I

2 solutions

Michael Mendrin
Dec 29, 2016

Given any two buildings, say, A and K, then any location for a community gas station will result in the same total sum of distances---provided it's between A and K. So, we want to find the building with an equal number of other buildings on either side of it. In this case, that would be building F.

If there were an even number of buildings then I suppose it would be a tie between the "middle two". For example, if building H were removed, then choosing either of buildings E or F would yield the same result.

Potential follow-up question: If we had a ring road, so that buildings A and K were now, say, 100 m apart (along the arc), would we still choose building F? (I think it might instead be building C we would choose in this scenario.)

Brian Charlesworth - 4 years, 5 months ago

Ossama Ismail
Dec 29, 2016

Just pick the $median$ of the buildings to be the gas station.

Why the median? Why not mean?

Pi Han Goh - 4 years, 5 months ago

Where is the gas station?

2 solutions

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