1. Optimization Problem (pumping station)

A river 200 km long, with Town A 90 km north of its west side, and Town B 60 km north of its east side. A water station must be built somewhere along the river so as to minimize the length of pipe needed to connect the station to both towns.

Find the position of the water station along the river.

Can anyone help me get started with this one? Thanks.

This just in: I found the answer to be 120 km along the river. If anyone can confirm if this is right, that would be great.

2. Originally Posted by Tulki
A river 200 km long, with Town A 90 km north of its west side, and Town B 60 km north of its east side. A water station must be built somewhere along the river so as to minimize the length of pipe needed to connect the station to both towns.

Find the position of the water station along the river.

Can anyone help me get started with this one? Thanks.

This just in: I found the answer to be 120 km along the river. If anyone can confirm if this is right, that would be great.
Let S be the point where the station is located.

The distance SA + SB will be minimized.

If the station is x units away from the west end, then it will be 200 - x units away from the east end.

Use the right triangles:

SA^2 = x^2 + 90^2

SB^2 = ( 200-x)^2 + 60^2

Solve for SA and SB above, respectively, then add both sides to get total length SA + SB = L(x)

Optimize L(x) = using calculus.

I hope this helps. Good luck!