An oil pipeline will connect points A and B which are 3 km apart and on opposite banks of a 1 km wide river. From point A to point C on the opposite bank, the pipeline will run under the water. From point C to point B the pipeline will run above ground.

The cost of running the pipeline under water is three times the cost of running it above ground.

Determine an equation that can be used to minimize the cost. You need not solve the problem.

Your language is ambiguous, but this kind of problem usually specifies that B is 3 km from a point O directly across the river from A.

If that is the case, then let OC=x and thus CB = 3-x
The cost function is then
f(x) = 3√(x^2+1) + (3-x)

If you meant to say that the line-of-sight distance AB=3, OB = √8, rather than 3.

To determine an equation that can be used to minimize the cost, we can define the variables involved in the problem:

Let:
x = the length of the pipeline running under water (in kilometers)
y = the length of the pipeline running above ground (in kilometers)

Based on the information given, we know that the total length of the pipeline from A to C to B is 3 km, and the width of the river is 1 km.

Since the pipeline runs under the water from A to C, the length of the pipeline under water is x. The length of the pipeline above ground from C to B is y.

From point A to point C, the length of the pipeline under water is x, and from point C to point B, the length of the pipeline above ground is y. Therefore, we have the equation:

x + y = 3

Additionally, the cost of running the pipeline under water is three times the cost of running it above ground. Assuming the cost of running the pipeline above ground is $c per kilometer, the cost of running the pipeline under water is $3c per kilometer. Therefore, the total cost can be expressed as:

Cost = 3c * x + c * y

We have now determined the equation that can be used to minimize the cost:

Cost = 3cx + cy

subject to the constraint:

x + y = 3

To determine an equation that can be used to minimize the cost, we need to express the total cost of the pipeline in terms of a single variable, which we can then minimize using calculus.

Let's denote the length of the pipeline that runs under water as x (in kilometers), and the length of the pipeline that runs above ground as (3 - x) kilometers.

The cost of running the pipeline under water is given as three times the cost of running it above ground. So, the cost of running the part of the pipeline under water is 3x, and the cost of running the part above ground is 1 * (3 - x) = 3 - x.

Therefore, the total cost of the pipeline is given by the sum of the costs of the two sections:

Total Cost = Cost of underwater section + Cost of above ground section
Total Cost = 3x + (3 - x)

Simplifying the equation, we get:

Total Cost = 2x + 3

Now we have an equation that represents the total cost of the pipeline in terms of the variable x. To minimize the cost, we can take the derivative of the equation with respect to x, set it equal to zero, and solve for x. This will give us the value of x at which the cost is minimized.

Finally, it is important to note that we have not solved the problem completely yet, as we have only determined the equation that can be used to minimize the cost.