Question

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 4 km east of the refinery. The cost of laying pipe is $200,000/km over land to a point P on the north bank and $400,000/km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located? (Round your answer to two decimal places.)

Answer 1

answer is 4000 - x meters
c = 2000/meter * (4000 - x) meters + 4000/meter [ sqrt (2000^2 + x^2)]meters

dc/dx = -2000 + 4000 [ .5] 2x / (x^2+4*10^6)^.5
for max or min dc/dx = 0
2000 (x^2+4*10^6)^.5 = 8000 x
(x^2+4*10^6)^.5 = 4 x
x^2 + 4*10^6 = 16 x^2
4 * 10^6 = 165 x^2
x^2 = 400 *10^4 /165 = 2.424242 ...... *10^4
x = 1.56 * 10^2 = 156
4000 - 156 = 3844 meters = 3.844 km
check my arithmetic !