A company wishes to build a pipeline from its oil rig which

[sh8485]

I am having difficulty understanding the following question

A company wishes to build a pipeline from its oil rig which is 30 km in the sea from the shoreline to a processing plant which is onshore and 20 km due east of the rig. It costs $50,000 to lay 1km of pipeline underwater and $20,000 to lay 1km of pipeline on land. What point on the shoreline must be selected to minimize the total cost of the pipeline?


Could you guys help me solve the problem?

Thanks

 

[mmm4444bot]

I'm thinking that it's a misstatement to say that the onshore plant is due east of the offshore rig (unless the shoreline is running something like SW to NE).

How about the following, instead?

The shoreline runs east to west.

The offshore rig is 30 km due north of a point on the shoreline.

The onshore plant is 20 km due east of this point on the shoreline.

(I'm drawing a picture, which I will post in a few minutes …)

 

[mmm4444bot]

(Arthur might be bummed over missing out because this is his favorite exercise.)

If necessary, double-click the image to expand it.

[attachment=0:2kj54o1r]drill-baby-drill.JPG[/attachment:2kj54o1r]

In this senario, you're asked to find some distance x, such that the total cost is minimized.

The red line segment represents underwater pipe; the green line segment represents pipe on land.

Write expressions in terms of x for each of these line segment's lengths.

Multiply the red length by 50000 (this product is the cost of laying the underwater pipe); multiply the green length by 20000 (this product is the cost of laying the pipe on land).

The sum of these two products defines the total cost function C(x).

Use C`(x) to find the value of x that minimizes C(x).

If you need more help, then please show us whatever work you can do, so that we might determine where to continue helping you.

 

[sh8485]

50,000(x^2+900)= 45,000,000+50,000x^2
20,000(20-x)= 400,000-20,000x

add two equation:

c(x)=45,000,000+50,000x^2+400,000-20,000x
c'(x)= 100,000x-20,000

 

[Deleted member 4993]

50,000(x^2+900)= 45,000,000+50,000x^2
20,000(20-x)= 400,000-20,000x

add two equation:

c(x)=45,000,000+50,000x^2+400,000-20,000x
c'(x)= 100,000x-20,000

Are you stuck here?

If you are - at the maximum value of a function (f(x)), what is the value of the derivative of the said function (f'(x))?

 

[mmm4444bot]

50,000(x^2+900) … There is an error, here.

The length of the underwater pipe is not x^2 + 900.

Did you use the Pythagorean Theorem?

Your expression for the cost of the pipe on the ground is good.