# An Oil Rig is located 10 miles from a straight coastline.?

An Oil Rig is located 10 miles from a straight coastline. An Oil Refinery is located 16 miles down the coastline. The cost of putting in a pipeline from the Oil Rig to the Oil Refinery depends on whether it is on land or in the water. The cost is $4000 per mile for land and $9000 per mile for water. How many miles of pipeline should be on land so that you are minimizing the cost of the entire pipeline?

### 1 Answer

- az_lenderLv 72 months ago
Let p be the number of miles of pipe that will be on land.

Then the number of miles of pipe in the water would be

sqrt[10^2 + (16 - p)^2].

The cost C in thousands of dollars is

4p + 9*sqrt[10^2 + (16-p)^2].

The derivative dC/dp is

4 + (9/2)[2(p-16]/sqrt[10^2 + (16-p)^2], or

4 + 9(p-16)/sqrt[10^2 + (16 - p)^2].

This will be zero when

144 - 9p = 4*sqrt[10^2 + (16 - p)^2], or

36 - (9/4)p = sqrt[10^2 + (16-p)^2], or

1296 - 162p + (81/16)p^2 = 100 + 256 - 32p + p^2 =>

(65/16)p^2 - 130p + 940 = 0.

Use the quadratic formula;

p = 16 +/- (8/65)*sqrt[130^2 - (65/4)(940)].

Only the "minus" makes sense, so

p = 16 - 4.96 = 11.04 miles.

To check for reasonableness, calculate

C(10), C(11), and C(12). I get

$144,957 and $144,623 and $144,932.

So the value p = 11 miles seems pretty good!

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