Extrema of a Cubic
What are the minimum and maximum values of the cubic polynomial function \({f(x) = x^3-6x^2+6x+6}\) restricted to the domain \({[0,5]}\)?
Goat Pen Along a River
Farmer Fabio wants to build a rectangular, fenced-in pen for his herd of goats along a riverbank; this way he only needs to build three sides of the pen and his goats will be free to frolic in the river shallows. He has 36′ of prefabricated fencing. What’s the maximum possible area of land along the river that the pen can enclose?
Pill-Shaped Container
Calypso is planning to 3D-print a large, hollow capsule in the shape of a cylinder with a hemispherical cap on one end, similar to a bullet cartridge (round) or an old grain silo. She only has enough filament to print a capsule with total surface area 932 in² though. What is the maximum possible volume of this capsule, and what dimensions attain this maximum?
Two Ships Passing in the Night
The routes of two cargo ships, the Ascent headed due north and the Briteaux heading due east, intersect in the middle of the Pacific Ocean. At the moment the ships first come within sight of each other, the Ascent is cruising towards the intersection point at a speed of 12 knots (nautical miles per hour) but is still 7 nautical miles from it, and the Briteaux is cruising towards the intersection point at a speed of 16 knots but is still 6 nautical miles from it. Eventually each ship passes the intersection and heads on its merry way. What is the nearest the ships ever get to each other?