Question

an offshore oil well is 5 kilometers off the coast. the refinery is 6 kilometers down the coast. laying pipe in the ocean is twice as expensive as on land. how many kilometers down the coast should the pipe be laid in order to minimize the cost?

Answer 1

The pipe should not be laid down the coast to minimize the cost.

Step-by-step explanation:

Let x represent the distance, in kilometers, down the coast where the pipe should be laid. The cost of laying the pipe in the ocean is twice as expensive as on land, so the cost of laying the pipe in the ocean is 2x times the cost of laying the pipe on land.

The total cost is obtained by adding the cost of laying the pipe in the ocean (2x) to the cost of laying the pipe on land (x). To find the value of x that minimizes the cost, we can take the derivative of the total cost equation with respect to x and set it equal to 0.

d/dx(2x + x) = 0

2 + 1 = 0

3 = 0

The derivative is a constant, which means that there is no value of x that minimizes the cost. Therefore, the pipe should not be laid down on the coast in order to minimize the cost.

Answer 2

To minimize the cost, the pipe should be laid 1 kilometer down the coast.

How to minimize cost

Given:

The offshore oil well is 5 kilometers off the coast.

The refinery is 6 kilometers down the coast.

Laying pipe in the ocean costs twice as much as on land.

We need to find the distance down the coast to lay the pipe in order to minimize the cost.

Let the distance the pipe is laid down the coast be denoted as 'x' kilometers.

The total cost for laying the pipe includes the cost of the offshore section and the cost of the on-land section.

If laying pipe in the ocean is twice as expensive as on land, then the cost per kilometer on land (C_land) can be compared to the cost in the ocean (2 * C_land).

Now, set up the cost equation:

Total cost (C) = Cost of offshore section + Cost of on-land section

The cost equation can be represented as:

C = 2 * C_land * 5 + C_land * x

C = 10 * C_land + C_land * x

C = C_land * (10 + x)

The total distance from the oil well to the refinery is 5 kilometers off the coast plus 'x' kilometers down the coast, which equals 6 kilometers.

Therefore, x = 6 - 5 = 1 kilometer

Hence, to minimize the cost, the pipe should be laid 1 kilometer down the coast.