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Mathematical modeling optimization problem
Solution:

Lay the pipeline as shown in the above picture.

Suppose: Point P is located at the downstream x(km) of the refinery, and 0≤x≤ 10. The total cost of laying is y billion yuan.

According to the meaning and known, there are:

y=4x+6√[2.5? +( 10-x)? ]

y=4x+6√(x? -20x+ 106.25)

y'=4+3(2x-20)/√(x? -20x+ 106.25)

y'=[4√(x? -20x+ 106.25)+6x-60]/√(x? -20x+ 106.25)

1, sequence: y' > 0, namely: [4√(x? -20x+ 106.25)+6x-60]/√(x? -20x+ 106.25)>0

Yes: 2√(x? -20x+ 106.25)+3x-30>0

30-3x