This problem is about minimizing the length of the road and bridge connecting two cities, while ensuring that the bridge is orthogonal to the river banks and crosses the river at its narrowest point.

Problem setup:

• One city is at point $A(-3, 4)$ and the other city is at point $B(8, -2)$.
• The river is centered along the $x$-axis and has width $w = 1$, meaning it extends from $y = -\frac{1}{2}$ to $y = \frac{1}{2}$.
• The bridge should be orthogonal to the river, implying that the bridge crosses the river along a vertical line at some $x = x_b$.

Objective:

Minimize the total length of the road, which consists of two segments on land and the bridge over the river.

1. Define the crossing point of the bridge:

Let the bridge be located at some point $x = x_b$. This means:

• The bridge starts at $(x_b, -\frac{1}{2})$ and ends at $(x_b, \frac{1}{2})$, with length 1 (since the river has width 1).

2. Define the road segments:

The road consists of two parts:

1. From city $A(-3, 4)$ to the bridge's start at $(x_b, \frac{1}{2})$.
2. From city $B(8, -2)$ to the bridge's end at $(x_b, -\frac{1}{2})$.

The total road length to minimize is: $L(x_b) = L_A(x_b) + L_B(x_b) + 1$ Where:

• $L_A(x_b)$ is the distance from $A(-3, 4)$ to $(x_b, \frac{1}{2})$,
• $L_B(x_b)$ is the distance from $B(8, -2)$ to $(x_b, -\frac{1}{2})$,
• 1 is the length of the bridge.

3. Express the distances:

1. The distance from city $A(-3, 4)$ to the bridge's start $(x_b, \frac{1}{2})$: $L_A(x_b) = \sqrt{(x_b + 3)^2 + \left(\frac{7}{2}\right)^2}$
2. The distance from city $B(8, -2)$ to the bridge's end $(x_b, -\frac{1}{2})$: $L_B(x_b) = \sqrt{(x_b - 8)^2 + \left(\frac{3}{2}\right)^2}$

4. Formulate the total length:

The total road length $L(x_b)$ is: $L(x_b) = \sqrt{(x_b + 3)^2 + \left(\frac{7}{2}\right)^2} + \sqrt{(x_b - 8)^2 + \left(\frac{3}{2}\right)^2} + 1$ Where $x_b$ is the variable representing the position of the bridge crossing the river.

5. Optimization:

We need to minimize $L(x_b)$ with respect to $x_b$. This is a standard optimization problem that can be solved using calculus:

• Take the derivative $\frac{dL}{dx_b}$,
• Set it equal to 0 to find the critical points,
• Verify that the critical point is a minimum by checking the second derivative or the behavior of $L(x_b)$.

Let me know if you'd like the next steps on solving this optimization problem.

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Questions to Expand Understanding:

1. How do we interpret the critical points of the function $L(x_b)$?
2. Why does the bridge need to be orthogonal to the river, and how does that affect the solution?
3. How would the solution change if the width of the river were different?
4. Can this method be extended to more complex terrain or city layouts?
5. What tools can we use to compute the derivative and solve the minimization problem?

Tip:

When solving optimization problems, always carefully define your objective function and constraints before jumping to calculations. This helps avoid errors and ensures clarity in the solution process.