To minimize the construction cost of the pipeline from $A$ to $B$, we need to determine the optimal location to cross the river, minimizing the total cost of both the water and land segments of the pipeline.

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Problem Setup

• The river is $0.75$ miles wide.
• $A$ and $B$ are on opposite banks, with $B$ located $2$ miles downstream from $A$.
• Cost per mile in water: $\$2 \, \text{million}$.
• Cost per mile on land: $\$1 \, \text{million}$.

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Approach

1. Variables: Let $x$ represent the distance downstream from $A$ where the pipeline crosses the river (the point where the pipeline exits the water).

2. Land Distance: The length of the pipeline on land is $2 - x$.

3. Water Distance: The length of the pipeline in water is $\sqrt{x^2 + 0.75^2}$ (using the Pythagorean theorem).

4. Cost Function: The total cost $C(x)$ in millions of dollars is: $C(x) = 2 \sqrt{x^2 + 0.75^2} + (2 - x)$ where $2 \sqrt{x^2 + 0.75^2}$ is the water cost, and $(2 - x)$ is the land cost.

5. Optimize: We minimize $C(x)$ by finding its derivative and solving $\frac{dC}{dx} = 0$.

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Steps

1. Derivative of $C(x)$: $\frac{dC}{dx} = \frac{4x}{\sqrt{x^2 + 0.75^2}} - 1$

2. Solve $\frac{dC}{dx} = 0$: $\frac{4x}{\sqrt{x^2 + 0.75^2}} = 1$ $4x = \sqrt{x^2 + 0.75^2}$ Square both sides: $16x^2 = x^2 + 0.75^2$ $15x^2 = 0.5625$ $x^2 = \frac{0.5625}{15}$ $x^2 = 0.0375, \quad x = \sqrt{0.0375} \approx 0.1938 \, \text{miles}.$

3. Compute $C(x)$:

   • Water distance: $\sqrt{0.1938^2 + 0.75^2} \approx 0.7755 \, \text{miles}.$
   • Land distance: $2 - 0.1938 = 1.8062 \, \text{miles}.$
   • Cost: $C(x) = 2(0.7755) + 1(1.8062) = 1.551 + 1.8062 \approx 3.3572 \, \text{million dollars}.$

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Final Answer

The cheapest construction cost is approximately $3.36 million.

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Would you like further clarification or details about the process?
Here are 5 related questions you might find interesting:

1. What happens if the river width changes?
2. How would the cost change if the water cost doubles?
3. Can we find the crossing point graphically instead of using calculus?
4. What if $B$ were further downstream?
5. How does the total cost change if both land and water costs are equal?

Tip: Drawing a diagram of the situation often helps clarify problems involving distances and geometry.