## A recurring strand in Exeter's math curriculum

Optimization problems arise in a variety of settings throughout our problems sets. Here are a few examples.

**Math 1**: Robin works part-time carving wooden seagulls and lobsters to sell to tourists during the summer season. Keeping up with demand means carving at least two seagulls and three lobsters a day. Robin can produce at most a dozen models a day. Can Robin make a profit of $70 in one day? What combination of seagulls and lobsters produces the greatest daily profit? What is this maximum profit?

(Math 1, #566)

**Math 2: **A spider lived in a room that measured 30 feet long by 12 feet wide by 12 feet high. One day, the spider spied an incapacitated fly across the room, and of course wanted to crawl to it as quickly as possible. The spider was on an end wall, one foot from the ceiling and six feet from each of the long walls. The fly was stuck one foot from the floor on the opposite wall, also midway between the two long walls. Knowing some geometry, the spider cleverly took the shortest possible route to the fly and ate it for lunch. How far did the spider crawl?

(Math 2, 24#2)

**Math 3: **The diameter and the slant height of a cone are both 24 cm. Find the radius of the largest sphere that can be placed inside the cone. (The sphere is therefore tangent to the base of the cone.) The sphere occupies a certain percentage of the cone’s volume. First estimate this percentage, then calculate it.

(Math 3, 29#5)

** Math 4: **Verify that the curve *y*^{2} + 2*x*^{2} = *x*^{3} + x is traced parametrically by *x* = *t*^{2} and *y* = *t*^{3} - *t*. Use these two equations to find the slope of the line that is tangent to the curve at (4, 6). Find coordinates for the two points on the curve where the *y*-coordinate is locally extreme.

(Math 4, 62#8, image from 62#3)