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A new cottage is built across the river and 300m downstream from the nearest telephone relay station. The river is 120m wide. In order to wire the cottage for phone service, wire will be laid across the river under water, and along the edge of the river above ground. The cost to lay wire under water is $15 per m and the cost to lay wire above ground is $10 per m. How much wire should be laid under water the minimize the cost?
And got the following answer:
let x be wire down the shore and C be the cost C = 10 (wire above water) + 15 (wire below water) C = 10x + 15√((120)² + (300-x)²) dC/dx = 10 - 15 (300 - x) / √((120)² + (300-x)²) 10 √((120)² + (300-x)²) = 15 (300 - x) .... set dC/dx = 0 to find stationary points 4((120)² + (300-x)²) = 9 (300-x)² 4(120)² = 5(300-x)² x = 12(25 - 4√5) ....... is an absolute min d = √((120)² + (300-(12(25 - 4√5)))²) = 72√5 Answer: put 72√5 m under water