Supertankers off-load oil at a docking facility shore point 4 miles offshore. the nearest refinery is 8 miles east of the docking facility. a pipeline must be constructed connecting the docking facility with the refinery. the pipeline costs $300,000 per mile if constructed underwater and $200,000 per mile if over land.

To minimize the total cost of constructing a pipeline from the docking facility to the refinery, we'll need to consider a path that involves both an underwater segment and a land segment.

### Problem Breakdown

- **Docking Facility (D):** Located 4 miles offshore.
- **Refinery (R):** Located 8 miles east of the docking facility.
- **Cost:** 
  - $300,000 per mile for underwater construction.
  - $200,000 per mile for land construction.

### Step-by-Step Solution

1. **Define Variables:**

   - Let $ x $ be the point on the shoreline directly east of the docking facility.
   - Let $ y $ be the point on the shoreline where the underwater pipeline meets the shore.
   - Let $ z $ be the distance from $ y $ to the refinery.

   Since the docking facility is 4 miles offshore, the distance between the docking facility (D) and $ y $ forms a right triangle.

2. **Calculate Distances:**

   - Underwater distance, $ D \to y $, is given by:
     \[
     d_u = \sqrt{4^2 + y^2} = \sqrt{16 + y^2}
     \]
   - Land distance from point $ y $ to the refinery $ R $ is:
     \[
     d_l = 8 - y
     \]

3. **Formulate the Cost Function:**

   The total cost $ C(y) $ of constructing the pipeline can be expressed as a function of $ y $:

   \[
   C(y) = 300,000 \times \sqrt{16 + y^2} + 200,000 \times (8 - y)
   \]

4. **Minimize the Cost Function:**

   To find the minimum cost, we need to find the derivative of $ C(y) $ with respect to $ y $ and set it to zero.

   \[
   \frac{dC}{dy} = 300,000 \times \frac{y}{\sqrt{16 + y^2}} - 200,000
   \]

   Set the derivative to zero to find the critical points:

   \[
   300,000 \times \frac{y}{\sqrt{16 + y^2}} = 200,000
   \]

   Simplifying,

   \[
   \frac{y}{\sqrt{16 + y^2}} = \frac{2}{3}
   \]

5. **Solve for $ y $:**

   Squaring both sides:

   \[
   \frac{y^2}{16 + y^2} = \frac{4}{9}
   \]

   Cross multiply:

   \[
   9y^2 = 4(16 + y^2)
   \]

   \[
   9y^2 = 64 + 4y^2
   \]

   \[
   5y^2 = 64
   \]

   \[
   y^2 = \frac{64}{5}
   \]

   \[
   y = \sqrt{\frac{64}{5}} = \frac{8}{\sqrt{5}} \approx 3.58 \text{ miles}
   \]

6. **Calculate the Minimum Cost:**

   Now, substitute $ y \approx 3.58 $ into the cost function to find the minimum cost.

   \[
   C(3.58) = 300,000 \times \sqrt{16 + (3.58)^2} + 200,000 \times (8 - 3.58)
   \]

Let's compute this value for a more precise result.The minimum cost to construct the pipeline is approximately **$2,494,427**.

### Summary:
- The optimal point on the shore where the pipeline meets the land is about 3.58 miles east of the point directly offshore from the docking facility.
- The total cost of the pipeline construction, using the most economical combination of underwater and land pipeline, is about $2,494,427.

Would you like more details on any step or have any further questions?

### Related Questions:
1. What is the impact on the cost if the underwater pipeline cost increases to $350,000 per mile?
2. How would the solution change if the refinery was not exactly 8 miles east but further inland?
3. Can we derive a general formula for any given distance and costs?
4. What would happen if the land and underwater pipeline costs were the same?
5. How sensitive is the total cost to changes in the value of $ y $?

**Tip:** When solving optimization problems, it is crucial to differentiate and find critical points that minimize or maximize the function in question.

Explore how to minimize the cost of constructing a pipeline from a docking facility 4 miles offshore to a refinery 8 miles east. Learn the steps involved in optimizing costs using both underwater and land construction methods. Estimated total cost: $2,494,427.

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