Question

An offshore oil well, P, is located in the ocean 5 km from the nearest point on the shore,A. A pipeline is to be built to take oil from P to a refinery that is 20km along the straight shore line from A. If it cost$100 000 per kilometre to lay pipe underwater and only $75 000 per kilometre to lay pipe on land, what route from the well to the refinery will be the cheapest?(answer to one decimal place)


The answer is 5.7 km but i got 14.33 km.

The equation i set up is:

c(x)=75000x+100000((20-x^2)+25)^(1/2)

Answer 1

c(x)=75000x+100000((20-x)^2+25)^(1/2)

sorry this is the equation i found.

Answer 2

OK. I will use 70 = 70k and 100=100k to get costs into $thousands.

Along the shore, you go toward point A a distance (20-x), at 75/km, and then cut through the water to P at 100/km. That slant path will have length of
sqrt[5^2 +x^2] = sqrt[25 + x^2]

Cost(x) = $75(20-x) + $100sqrt[(25+x^2)]= 1500-75x + 100[25+x^2]^1/2
dC/dx = 0 = -75 + 100(1/2)(2x)[25+x^2]^-1/2 = -75 + 100 x [25+x^2]^-1/2
0.75 = x /[25+x^2]^1/2 square both sides
0.5625 [25 + x^2] = x^2
14.06 = (1-0.5625) x^2
x^2 = 32.14
x = 5.7 whew! QED

hope that helps

Answer 3

what if we instead approached R from P? Would that work? @douglaswinslowcooper