An oil refinery is located on the north bank of a straight river that is 2 km2 km wide

The expression for length from point P to storage tank is,

l1=√22+P2=√4+P2l1=22+P2=4+P2

The expression for the length of the refinery to the point P is,

l2=6?Pl2=6?P

The cost of the pipe on the land is,

Cs=cs×l2Cs=cs×l2

Substitute the given values.

Cs=400000×(6−P)Cs=400000×(6−P)

The cost of the pipe in the water is,

CW=cW×l2CW=cW×l2

Substitute the given values.

Cs=800000×√4+P2Cs=800000×4+P2

The expression for the total cost is,

TC=400000×(6−P)+800000×√4+P2TC=400000×(6−P)+800000×4+P2

Differentiate the above function and equate to zero for the maximum value of P is,

−400000+12×800000(4+P2)−1/2⋅2P=0√4+P2=2P4+P2=4P2P2=43P=2√3=1.154−400000+12×800000(4+P2)−1/2⋅2P=04+P2=2P4+P2=4P2P2=43P=23=1.154

Substitute the given values.

TC=400000×(6−2√3)+800000×√4+43=$2540341.184TC=400000×(6−23)+800000×4+43=$2540341.184

Thus the minimum cost to lay the pipeline is $2540341.184$2540341.184.

Total Cost

The total cost of the company can be of two types fixed coat and variable value. All those costs which require one time at the time of initializing the organization are termed as a fixed cost, and that cost which varies with time is a variable.