Calculus Optimization Problem

Question

Optimization Problem. a home in the woods needs electricity. it costs $400 per mile to build on the road and $550 per mile to build in the forest. the home is 7 miles directly out int forest from a point that is 22 miles on the road from a substation. Determine the cost to go from the substation straight along the road(22 miles) then through the forest(7 miles) to the house. Determine the cost to run the electricity entirely to through the forest. Determine the cost to go partway on the road and partway through the forest. Use calculus to find the x that minimizes the cost along this path. How far along the road should we run the electricity before turning and going through the woods and which path we should use to run the electricity

Answer 1

Cost through 22 miles of road then 7 miles of forest:

 

22(400) + 7(550) = $12,650

 

 

Cost through √(22² + 7²) = 23.1 miles of forest from substation:

 

23.1(550) = $12,698

 

 

Cost through x miles along the road and √[7² + (22 - x)²] miles through the forest:

 

C(x) = 400x + 550√[7² + (22 - x)²]

 

C' = 400 - 550[7² + (22 - x)²]^-1/2(22 - x)

 

 

Setting C' = 0:

 

(8/11)²[7² + (22 - x)²] = (22 - x)²

 

49(8/11)² = (57/121)(22 - x)²

 

x = 14.6 miles

 

C(14.6) = $9,970