Distance:

Find the point on the plane $x-y+z = 4$ that is closest to the point $(1,2,3)$.

Find the points on the surface $y^2 = 9+xz$ that are closest to the origin.

Boxes!

Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm$^2$.

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant $c$.

If the length of the diagonal of a rectangular box must be $L$, what is the largest possible volume?

Other Optimization Problems

Three alleles (alternative versions of a gene) A, B, and O determine the four blood types A (AA or AO), B (BB or BO), O (OO), and AB. The Hardy-Weinberg Law states that the proportion of individuals in a population who carry two different alleles is $$P = 2pq + 2pr + 2rq$$ where $p,q,$ and $r$ are the proportions of A, B, and O in the population. Use the fact that $p+q+r = 1$ to show that $P$ is at most $2/3$.

Find an equation of the plane that passes through the point $(1,2,3)$ and cuts off the smallest volume in the first octant.