Analytical Approach

To analytically solve this problem, we are going to use the method of constrained optimization via subsitution. The general premise of this is instead of using external functions as our constraints when optimizing a problem, we can instead substitute the constrains into the equation and then optimize. Solving for the optimal (minimizing) perimeter involves somehow substituting the constraint for area into the perimeter equation before optimizing. Solving for the optimal (minimizing) area involves subsituting the constraint of perimeter into the equation before optimizing. Optimizing both area and perimeter give us the optimal relationship between base and height, not a physical value. This optimized relationship (e.g. B = 2*H) is then subsituted into the flow rate equation to find the actual value for one of the variables that makes the flow rate function true. In effect, we are finding the optimized ratio between base and height using the perimeter and area equations that will allow us to determine the smallest possible values for base and height that satisfy the flow rate requirements.

Once we have determined the correct values for base and beight (in meters) from the flow rate equation, we can plug these into our cost equation to find the minimum cost. Remember, we do not need to optimize the cost function using this approach because we have already minimized base and height to give us the smallest possible area and perimeter that still satisfy the constraint of the flow rate equation. In effect, area and perimeter act as constraints for each other to find the optimal ratio of base to height which, when substituted into the flow rate equation, minimizes the final numerical values for B and H. These final numerical values represent the minimum base and height required that satisfy the flow rate constraint, area constraint, and perimeter constraint. Because we already minimized area and perimeter, we can easily determine the cost without needing to take any additional derivatives.

The good news is, no further complex optimization is required! We know this by looking at the equation for cost above. The output "C" is determined by two inputs, the area and the perimeter of the canal. Most people's first guess would be to plug in equations 3 and 4 for area and perimeter into the above equation and optimize in terms of B and H. This approach, however, is incorrect. This is because it does not take into account the constraints on A and P (namely their relationship), and it also does not take into account the flow rate requirements.

The tricky yet convenient part of this is, we already know what the optimized relationships between A and P are with the constraints taken into account. This is because we found the relationship between B and H that minimized perimeter while also satisfying the flow rate requirements. We then used this relationship between B and H to determine the smallest cross-sectional area that satisfied the flow rate requirements, which is seen in equation 5 (told you we would come back to it!). There is another way to determine the minimized area that satisfies the perimeter constraint. We could use the same substitution method for constrained optimization but substitute the perimeter constraint into the differential equation that optimizes area (see the math in appendix II). This will give us the same result A = 2*(H^2 ) and P = 4H with a ratio of B = 2H. Because we are not solving for the actual base and height values, just the relationship between them that minimizes A and P, we can then plug our result into the flow rate equation to determine the numerical length based on the given Q, s, and n values,

Looking back at the equation above, we can see how minimizing the A term would bring the c1*A term to its smallest possible value, and minimizing P would also minimize the c2*P term (since c1 and c2 are constants). We cannot, however, let the "A" input approach any number it wants. It needs to satisfy the perimeter constraint, as the two terms are connected by the mutual sharing of B and H terms. This is also true in reverse, where the P term must only hold values that satisfy the relationship between A and P.

We cannot let both terms approach zero either, as at some point the area would get small enough torestrict flow below our flow rate requirements (acting as another constraint to the cost function).

Again, fortunately we already did all of this work. We already found the exact base and height dimensions that minimize perimeter while creating a canal that satisfies the flow rate requirements and the area-to-perimeter relationship. Using these known values, we can simply plug them into the cost equation to determine the smallest possible cost.

This may be in contrast with intuition. Given the fact that the cost of excavation is different from the cost of lining, why isn't there a new "optimized" base and height combination that minimize cost? Despite this reasoning, the answer is that regardless of what c1 and c2 are, we already found the smallest possible area that satisfies the flow rate, and the smallest perimeter that satisfies this area (and subsequently the smallest perimeter that satisfies the flow rate). Flow rate is particularly dependent on area, as we can see from the simplified equation with area and perimeter on one side. An increase in perimeter (holding area constant, as well as "n" and "s") would necessarily mean a less efficient shape (requiring excessive lining per unit area and non-optimized flow).