A company needs to run an oil pipeline from an oil rig 25 miles out to sea to a storage tank that is 5 miles inland. The shoreline runs east-west and the tank is 8 miles east of the rig. Assume it costs $50 thousand per mile to construct the pipeline under water an $20 thousand per mile to construct the pipeline on land. The pipeline will be built in a straight line from the rig to a selected point on the shoreline, then in a straight line to the storage tank. What point on the shoreline should be selected to minimize the total cost of the pipeline?
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D_water = √(x² + 25²)
D_land = √((8 - x)² + 5²)
Cost = $_water*D_water + $_land*D_land
C(x) = 50√(x² + 25²) + 20√((8 - x)² + 5²)
C(x) = 50√(x² + 625) + 20√(x² - 16x + 89)
C'(x) = 50x/√(x² + 625) + 20(x - 8)/√(x² - 16x + 89) = 0
...
x ≈ 5.10898675360722819945065016856194924837…
(8 - x) ≈ 2.89101324639277180054934983143805075162…
C(5.1899) ≈ $1.391 Million
D_land = √((8 - x)² + 5²)
Cost = $_water*D_water + $_land*D_land
C(x) = 50√(x² + 25²) + 20√((8 - x)² + 5²)
C(x) = 50√(x² + 625) + 20√(x² - 16x + 89)
C'(x) = 50x/√(x² + 625) + 20(x - 8)/√(x² - 16x + 89) = 0
...
x ≈ 5.10898675360722819945065016856194924837…
(8 - x) ≈ 2.89101324639277180054934983143805075162…
C(5.1899) ≈ $1.391 Million