To solve this problem, let's break down the setup and construct the required expressions.

Step 1: Understand the Geometry and Setup

We have:

1. Points $B$ and $C$ are 4 km apart.
2. Point $A$ is 4 km from the midpoint of $B$ and $C$.
3. A point $D$ lies on the line segment between $B$ and $C$, at a distance $h$ from the midpoint of $B$ and $C$, where we need to minimize the total distance of digging.

We want to find the minimum length of the ditch, which involves digging from $A$ to $D$ and from $D$ to both $B$ and $C$.

Step 2: Define the Distance Expressions

1. Distance from $A$ to $D$: This is a vertical distance of $4 + h$ km, given that $A$ is vertically below $D$ by 4 km and $D$ is displaced by $h$ km from the midpoint of $B$ and $C$.

2. Distances from $D$ to $B$ and $C$: Since $B$ and $C$ are equidistant from the midpoint, the distance from $D$ to $B$ and $C$ forms two identical right triangles with legs $h$ and $2$ (half of the 4 km between $B$ and $C$).

   The length from $D$ to $B$ or $C$ is therefore: $\sqrt{h^2 + 2^2} = \sqrt{h^2 + 4}$

3. Total Length $L(h)$ of Ditches: The total ditch length $L(h)$ is the sum of the distances from $A$ to $D$ and from $D$ to both $B$ and $C$ (since both are symmetric).

   $L(h) = (4 + h) + 2\sqrt{h^2 + 4}$

Step 3: Minimize $L(h)$

To find the value of $h$ that minimizes $L(h)$, we need to take the derivative of $L(h)$ with respect to $h$, set it to zero, and solve for $h$.

1. Differentiate $L(h)$ with respect to $h$: $\frac{dL}{dh} = 1 + 2 \cdot \frac{h}{\sqrt{h^2 + 4}}$

2. Set the derivative equal to zero to find critical points: $1 + 2 \cdot \frac{h}{\sqrt{h^2 + 4}} = 0$

   Solving this equation will give us the value(s) of $h$ that minimize the ditch length.

Step 4: Solve for $h$

To simplify, let’s proceed with solving the above expression. I’ll perform the calculations now.The solution for $h$ that minimizes the total length of the ditch is:

$h = -\frac{2\sqrt{3}}{3} \approx -1.15 \text{ km}$

Since $h$ represents a physical distance (from the midpoint of $B$ and $C$ towards $A$), a negative value here implies moving below the midpoint, which may not align with the given setup if $h$ must be a positive distance above the midpoint.

Thus, this solution suggests that to achieve minimal digging, point $D$ might need to be below the midpoint by $\frac{2\sqrt{3}}{3}$ km. However, if we restrict $h$ to positive values only, further considerations or constraints may be needed.

Would you like further details on how this affects the ditch length calculation or any clarification?

Follow-up Questions

1. How would restricting $h$ to positive values alter the solution?
2. What is the total minimum ditch length if $h$ is negative?
3. How does the geometry change if $D$ is allowed to go below the midpoint?
4. Is it possible to minimize the ditch length under a constraint that $h \geq 0$?
5. How would the solution differ if the distance from $B$ to $C$ was different?

Tip:

In optimization problems, always verify if the solution meets all physical or practical constraints in the given context.