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Do the mathematical formulation and the problem solution based on the rubric and the sample sba. **Define Variables:**

Let ‘x’ be the Distance from **B** (directly onshore from **A**) to **D** (where the pipeline meets the shore).

Let ‘y’ be the Length of the underwater pipeline (**AB**).

Let ‘z’ be the Length of the onshore pipeline (**DC**).

Let ‘p’ be the Cost per km of onshore pipeline.

Let ‘q’ be the Cost per km of onshore pipeline.

Given that **AB = 7 km**, **BC = 10 km**.

### Mathematical Formulation:

**Variables:**
- Let \( x \) be the distance from point **B** (onshore, directly from **A**) to **D** (the point where the pipeline meets the shore).
- Let \( y \) be the length of the underwater pipeline (**AB**).
- Let \( z \) be the length of the onshore pipeline (**DC**).
- Let \( p \) be the cost per kilometer for the onshore pipeline.
- Let \( q \) be the cost per kilometer for the underwater pipeline.

**Given:**
- \( AB = 7 \, \text{km} \) (length of the underwater pipeline),
- \( BC = 10 \, \text{km} \) (distance from the shoreline to the oil storage facility).

---

### Cost Formulation:

**Total Cost (C):**

We need to find the total cost of running the pipeline. The total cost is the sum of the costs of the underwater pipeline and the onshore pipeline:
\[
C = p \cdot z + q \cdot y
\]
where:
- \( z \) is the length of the onshore pipeline from **D** to **C**.
- \( y \) is the length of the underwater pipeline from **A** to **B**.
- \( p \) is the cost per kilometer to run the pipeline onshore.
- \( q \) is the cost per kilometer to run the pipeline underwater.

**Relationship between Variables:**

Since \( \triangle ABC \) is a right triangle, we can apply the Pythagorean theorem to determine the relationship between \( x \), \( y \), and \( z \).

From the given, we know that the straight distance \( BC = 10 \, \text{km} \). Also, \( AB = 7 \, \text{km} \), and \( B \) is perpendicular to **AC** at point **B**, which is directly onshore from **A**.

Using the Pythagorean theorem:
\[
AB^2 + BC^2 = AC^2
\]
\[
7^2 + x^2 = 10^2
\]
\[
49 + x^2 = 100
\]
\[
x^2 = 51
\]
\[
x = \sqrt{51} \approx 7.14 \, \text{km}
\]

This means that **x**, the distance from **B** to **D**, is approximately \( 7.14 \, \text{km} \).

So, **z** (the length of the onshore pipeline from **D** to **C**) will be the remaining part of the total distance from **B** to **C**, i.e.:
\[
z = BC - x = 10 - 7.14 = 2.86 \, \text{km}
\]

---

### Problem Solution:

The next step is to determine the total cost for both companies and decide which company should be awarded the contract.

#### Company A:
- \( p = 415,000 \, \text{USD/km} \) (cost per kilometer for onshore pipeline),
- \( q = 635,000 \, \text{USD/km} \) (cost per kilometer for underwater pipeline).

#### Company B:
- \( p = 390,000 \, \text{USD/km} \),
- \( q = 650,000 \, \text{USD/km} \).

The total cost for both companies is calculated by substituting the appropriate values into the total cost formula:
\[
C = p \cdot z + q \cdot y
\]

**For Company A:**
\[
C_A = 415,000 \cdot 2.86 + 635,000 \cdot 7
\]
\[
C_A = 1,187,900 + 4,445,000
\]
\[
C_A = 5,632,900 \, \text{USD}
\]

**For Company B:**
\[
C_B = 390,000 \cdot 2.86 + 650,000 \cdot 7
\]
\[
C_B = 1,114,000 + 4,550,000
\]
\[
C_B = 5,664,000 \, \text{USD}
\]

---

### Conclusion:
Company A should be selected for the contract because their total cost is less than that of Company B.

Let me know if you'd like more detailed explanations or have other questions! 

Here are five follow-up questions:
1. How do you find the minimum cost if the underwater pipeline’s length changes?
2. Can you explain how the Pythagorean theorem applies here?
3. How can you modify the model for a more complex pipeline scenario?
4. What is the significance of cost optimization in real-world engineering projects?
5. How does the distance between the points affect the total cost?

**Tip**: Always ensure to clearly define your variables at the start of any modeling problem to avoid confusion later on.

Do the mathematical formulation and the problem solution based on the rubric and the sample SBA. Define Variables: Let ‘x’ be the Distance from B (directly onshore from A) to D (where the pipeline meets the shore). Let ‘y’ be the Length of the underwater pipeline (AB). Let ‘z’ be the Length of the onshore pipeline (DC). Let ‘p’ be the Cost per km of onshore pipeline. Let ‘q’ be the Cost per km of onshore pipeline. Given that AB = 7 km, BC = 10 km.

Explore how to model the cost of laying a pipeline using mathematical formulas and cost optimization techniques. This detailed solution involves the application of the Pythagorean Theorem and algebraic expressions to find the minimum cost for two companies bidding to lay the pipeline. Given data includes 7 km of underwater pipeline and a 10 km shoreline distance.

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