7.5: Summary

1. Optimization is a process of finding critical points, and identifying local and global maxima/minima.
2. A scientific problem that address "biggest/smallest, best, most efficient" is often reducible to an optimization problem.
3. As with all mathematical models, translating scientific observations and reasonable assumptions into mathematical terms is an important first step.
4. The following applications were considered:
   1. Density dependent population growth. Using a given logistic growth law, the following parameters were considered:
      • population growth rate (to be maximized),
      • population density,
      • intrinsic growth rate (constant),
      • carrying capacity (constant).
   2. Nutrient absorption in a cell. Using the model developed in Section $1.2$ for a spherical cell, we considered:
      • nutrient absorption rate,
      • nutrient consumption rate,
      • cell radius,
      • proportionality constants (determined based on context). We maximized the net rate of increase of nutrients - a difference between absorption and consumption rates.
   3. Surface area of a cylindrical cell, which tends to be minimized do to energy conditions. The parameters we used were:
      • cell length,
      • cell radius,
      • cell volume (constant),
      • cell surface area (to be minimized).
   4. Wine for Kepler’s wedding, seeking the largest barrel volume for a fixed diagonal length. The following parameters were considered:
      • barrel volume, (to be maximized)
      • barrel height,
      • barrel radius,
      • length of the diagonal (constant).
   5. Foraging time for an animal collecting food. We considered:
      • travel time between nest and food patch,
      • foraging time in the patch,
      • energy gained by foraging in a patch for various time durations.

Quick Concept Checks

1. If the growth rate of a population follows the following logistic equation:

$$G(N)=1.2 N\left(\frac{50000-N}{50000}\right), \nonumber$$

where $N$ is the density of the population, under what circumstances is the population growing fastest?

2. When finding a global maximum, why is always imperative to check the endpoints?
3. Demonstrate the variability of barrel dimensions by giving two different height and radius pairs which lead to a volume of $50 \mathrm{~L}$.
4. Would the answer to Kepler’s wine barrel problem have changed if we had solved for $h^{2}$ instead of $r^{2}$ ?