Last week the students in my College Algebra started exponential functions. They got started on the basics of exponential graphs, when they are increasing and decreasing, and exponentials with a base e. With these basics completed, they are working on an Applications Quiz this week. To get them started on this material, I introduced an application based on Gilligan’s Island.

Put a bunch of people of both genders on a deserted island…what should happen?

Those of you who are fans of classic TV will recognize this as introduction to the sitcom “Gilligan’s Island”. This series ran from September 26, 1964 to September 4, 1967. The series revolved around 7 castaways (4 men and 3 women) marooned on an island somewhere near Hawaii.

Over the course of three seasons, the series followed attempts to leave the island, visitors to the island and general incompetence in getting rescued. When the series was cancelled in 1967, the castaways were never rescued.

In 1978, a made for TV movie called “Rescue from Gilligan’s Island” aired in which the castaways were rescued. However, at the end of this movie they decided to go on a reunion cruise and became stranded on the same island again after another freak storm. In 1979, another made for TV movie called “The Castaways on Gilligan’s Island” aired in which they were rescued once again. This time they decide to convert the island to a resort. It was hoped that this premise would generate a new series, but this never happened.

In 1981, a second sequel was created, “The Harlem Globetrotters on Gilligan’s Island” in which nefarious forces plotted to take over the island and the castaways are saved by the Harlem Globetrotter.

Although this series began before I was born, I watched years and years of reruns throughout my childhood. I must have watched each of the 98 episodes several times each. But there was one question that nagged at me. If there were two women on the island of child bearing age (Ginger and Mary Ann), why didn’t the population grow? Why didn’t nature take its course and lead to new castaways?

**Let’s find out how many castaways there should have been in 1978, 14 years after they were originally shipwrecked.**

To answer this question, let’s assume that the population on the island is an example of continuous exponential growth. This may or may not be a good choice due to the small size of the population. However, with this assumption the population after t years 1964 will be given by the function

In this situation, P is the initial population and r is the continuous growth rate in percent per year.

Since the population of Gilligan’s Island was initially 7, we’ll use a fairly conservative rate of 2% per year. This is about what the world birth rate is. Around the world birth rates vary from a little less than 1% (China) to around 5% (African countries). With these values, we model the population by

To predict the population in 1978, I’ll graph this function and find its value at *x* = 14.

According to this growth model, there should have been about 9 people on the island in 1978. Even though they had plenty of food and visitors, the population remained fixed throughout the series. The permanent population never reached 9 as continuous population growth of 2% would predict.

One can only imagine what would happen now. To compete with reality TV shows like “Survivor”, a remake would have Gilligan much more handsome. Mary Ann would be pregnant. The Skipper would have a drinking problem. The Professor would have episodes of schizophrenia. And can you imagine Mr. Howell on Viagra?

After all of this time, I have to think that even Gilligan would start to look good. Maybe then the population might start doubling as continuous exponential growth would dictate.