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A construction crew has to dig ditches for a set of pipes that must carry water from point A to both of points B and C (see figure). Points B and C are 4km
 apart and point A is 4km
 from the midpoint of B and C. They want to dig as little as possible and they believe the best way to do that is to dig from point A to some point D along the midline a distance h
 from the midpoint of B and C. They are not sure exactly what value of 
 to use. Formulate an expression L(h)
 for the total length of ditch. Now calculate the value of h
 which minimizes the amount of ditch to be dug. If there is more than one such value, give your answer as a comma-separated list.

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.

A construction crew has to dig ditches for a set of pipes that must carry water from point A to both of points B and C (see figure). Points B and C are 4 km apart and point A is 4 km from the midpoint of B and C. They want to dig as little as possible and they believe the best way to do that is to dig from point A to some point D along the midline a distance h from the midpoint of B and C. They are not sure exactly what value of h to use. Formulate an expression L(h) for the total length of ditch. Now calculate the value of h which minimizes the amount of ditch to be dug. If there is more than one such value, give your answer as a comma-separated list.

Explore a geometry and optimization problem where a construction crew minimizes the total length of ditch required to connect points A, B, and C. The solution involves formulating an expression L(h) for the total length and using derivatives to find the optimal distance h from the midpoint. Suitable for students in Grade 12 or college level.

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