Early in $1492$, Crist\'{o}bal Col\'{o}n was commissioned by King
Ferdinand and Queen
Isabella of Spain to journey west to reach the Orient. On September 6, 1492,
Columbus (Col\'{o}n's name in Latin) left the Canary
Islands to make history. Not
knowing exactly which direction he should head, Columbus manages to get his
ship moving east or west according to the function $x(t)=(t-36)^3 (100-t)$.
We pick up the action at $t=0$ and follow him as $t\to\infty$. (Note that
$t$ is in days, $x(t)$ is in meters, and that we ignore the reality
that Columbus sailed southwestward, not just east and west.)
- Assuming September 6, 1492 is the day we start watching,
what is the position of the Canary Islands?
- Since we are given that Columbus is initially travelling west, what is your sign convention?
- For which $t's$ is Columbus' ship stopped?
- When is he moving west? east?
- There is an island at $x=0$. How many times does Columbus go past this island? When?
- When did Columbus stop at this island, called Guanahani by its natives,
which he claimed for Spain and renamed San Salvador? From this, can you calculate the date of this reknowned stop?
- At $t=67\frac{3}{ 4}$ days, Columbus realizes that his second-in-command,
Pinz\'{o}n, and the Pinta had vanished before a strong east wind. At this point, Columbus thought that it would be a good idea to go east to find his friend.
When (for which time $t$) does the ship begin to slow down? What was the reaction time from seeing
this to slowing down? How far did he travel this time?
- For which $t$ is he:
- speeding up in the westward direction?
- slowing down in the westward direction?
- speeding up in the eastward direction?
- slowing down in the eastward direction?
- What is his position when he finally starts going east toward the island
of Haiti which he named Espa\~{n}ola (Hispaniola in Latin)?
- How many days elapse before Columbus sets his sights upon San Salvador
once again?
- Columbus is now headed back to Spain. At what time $t$ on his way to Spain are we sure that Columbus has passed the Canary Islands?
- How can we alter our function $x(t)=(t-36)^3 (100-t)$ so that Columbus
is allowed to stop at Espa\~{n}ola and board the Ni\~{n}a since the Santa
Maria ran aground at Espa\~{n}ola and was a total loss? Furthermore, how can
we ensure (by altering the function $x(t)$) that Columbus eventually stops as
$t\to \infty$? Keep in mind that after his stop in Haiti, Columbus must travel east.
Tags: monotonicity, finding-roots, critical-points, velocity, physics-application, acceleration, limits, limits-at-infinity, curve-sketching, geometry, higher-polynomial, word-problem, history, multistep, computation, v--a, t2, c3
This fascinating problem was researched and written by James Epperson (Thanks James!) Expect students to have a difficult time staying interested in such a long problem. You may want to counter this by splitting up the problem, and having different groups work on different parts. Another option is to offer extra credit for finishing the worksheet outside of class, and turning in a nicely written solution. Ideally, you can time it so that this worksheet appears around Columbus Day, but that’s not always possible (especially in the spring semester!) Be aware of the many different perspectives your students might have on Columbus (oppressor of my people, colonialist bastard, discoverer of my great country, etc.)
– Dave Kung 1998-08-09