To solve the problem of determining the maximum volume of the canal, we need to calculate the volume of the canal by considering its cross-sectional area and length. Here's the step-by-step process:

Step 1: Cross-sectional area of the isosceles trapezoid

The area $A$ of a trapezoid is given by: $A = \frac{1}{2} (a + b) y$ where:

• $a = 4 \, \text{m}$ (the shorter base),
• $b = 6 \, \text{m}$ (the longer base),
• $y$ (the height of the trapezoid).

From the diagram, $y$ depends on the slope $\theta$ of the sides, so: $y = x \tan(\theta)$ where $x$ is half the horizontal difference between $b$ and $a$, given by: $x = \frac{b - a}{2} = \frac{6 - 4}{2} = 1 \, \text{m}.$

Thus: $y = \tan(\theta).$

Step 2: Volume of the canal

The canal has a length of $L = 200 \, \text{m}$. Its volume $V$ is: $V = A \cdot L = \frac{1}{2} (a + b) y \cdot L.$

Substituting $y = \tan(\theta)$: $V = \frac{1}{2} (a + b) \tan(\theta) \cdot L.$

Substituting the known values $a = 4$, $b = 6$, and $L = 200$: $V = \frac{1}{2} (4 + 6) \tan(\theta) \cdot 200.$ $V = 1000 \tan(\theta).$

Step 3: Maximize the volume

To maximize $V$, we need to find the angle $\theta$ that maximizes $\tan(\theta)$, under the constraint that the trapezoid remains physically possible. The maximum feasible $\theta$ depends on practical design limits (e.g., the slope of the sides).

Result:

The volume $V$ depends directly on $\tan(\theta)$, and maximizing $\tan(\theta)$ maximizes $V$. If there's a specific angle constraint or other details you'd like analyzed, please clarify, and I can refine the solution!

Would you like further clarifications or an extended explanation?

Here are 5 related questions for deeper understanding:

1. What practical constraints might limit the angle $\theta$ in the construction of the canal?
2. How does the volume change if the length $L$ of the canal is reduced?
3. What happens to the volume if the width $b$ of the top base is increased?
4. How does the design of the canal affect water flow efficiency?
5. Can you analyze the structural stability of the canal for different slopes $\theta$?

Tip: Always check feasibility constraints (like slope and material strength) when maximizing geometric quantities!