"Much of the complex art of engineering is the art of optimization." - American mathematician Richard Hamming
The situation arises often. In fact, as suggested by the quote above, almost every engineering endeavour has the same central theme: to get "the most" out of some process or product usually within the context of various competing interests. We want the most power for the least gas consumption, we want the tallest structure with the least amount of materials, we want the most widgets in the least amount of time. This is the art of optimization, finding the "best" balance. Making a powerful engine is fairly straightforward, but few consumers are interested in filling the tank every 100 km!
A big part of the job as an engineer is working out exactly what we want and how to balance the variables at play in some ideal sense. And this must often be done in the face of very real constraints related to e.g. material properties, scheduling issues, market factors, budgets, government regulations, etc. While this is the reality of engineering work, we will not concern ourselves with such issues or constraints. Mathematically speaking, accounting for them is actually quite tricky. For our work, we will assume that:
In this light, our optimization problems can be discussed purely in mathematical terms.