optimization?

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 per km over land to a point P on the north bank and $400,000 per km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located? (Give your answer correct to two decimal places.)

Answer 1

Hi,

Let x be the distance from the oil refinery along the river to bury pipe underground. Then the remaining diagonal distance to lay pipe under the river would be found by finding the hypotenuse of the right triangle formed by the legs of the remaining 4 - x km parallel to the river bank and the 2 km leg across the river's width. That hypotenuse for the underwater distance would be √((4 - x)² + 2²). It costs $200,000 per km over land and $400,000 per km under the river. The formula to find the cost for the pipe with x km on land is:
...........................................__________
Y1 = 200000x +400000√((4 - x)² + 2²)

If you graph this in an appropriate window, you can use the calculator's minimum command to find what x value minimizes the cost. That occurs when x = 2.845 km buried on land before going diagonally across to the storage tanks.

In this case, the total expense is $1,492,820.32.

I hope that helps!! :-)

Answer 2

Call x the distance in east direction of P to the refinery and y
the length of the river crossing
Cost =200,000*x +400,000*y
You have to find the minimum of x+2y with the condition
(x-4)^2+2^2-y^2 =0 (Pythagoras) (a)
Usin lagrangian multipliers
F(x,y,k) = x+2y+k[(x-4)^2+4-y^2]
Partial derivates
Fx= 1+2k(x-4)=0
Fy= 2-2ky=0
so y=1/k and x= (8k-1)/2k
going to (a)
1/4k^2+4-1/k^2 =0 so 3/4k^2 =4 so k^2 =3/16 and k = 1/4*sqrt3 (+ sign as y must be posirive)
so x= (12-2sqrt3)/3 =4-2/3*sqrt3= 2.845 Km rounded to 2.85km
Please check calculations

Answer 3

If P is 2 km east of the refinery on the north bank and then the pipe goes across the river to the storage tanks, the cost will be 1,531,370.85