7.5: Summary
- Page ID
- 121204
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- Optimization is a process of finding critical points, and identifying local and global maxima/minima.
- A scientific problem that address "biggest/smallest, best, most efficient" is often reducible to an optimization problem.
- As with all mathematical models, translating scientific observations and reasonable assumptions into mathematical terms is an important first step.
- The following applications were considered:
- 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).
- 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.
- 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).
- 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).
- 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.
- Density dependent population growth. Using a given logistic growth law, the following parameters were considered:
- 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?
- When finding a global maximum, why is always imperative to check the endpoints?
- Demonstrate the variability of barrel dimensions by giving two different height and radius pairs which lead to a volume of \(50 \mathrm{~L}\).
- Would the answer to Kepler’s wine barrel problem have changed if we had solved for \(h^{2}\) instead of \(r^{2}\) ?