Doubling Time on Gilligan’s Island

Populations are often described by their doubling times. The doubling time is the amount of time it takes a population to double. We can calculate the doubling time of an increasing exponential model by examining its graph carefully. Since students seem to enjoy themodel for population growth on Gilligan’s Island, I like to use it to illustrate doubling in an exponential function. Now that we have looked at a model for Gilligan’s Island, let’s see what its doubling time is. Recall that the amount of people on Gilligan’s Island t years after 1964 may be modeled by the function A(t) where

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and the population is growing by 2% continuously. Since Gilligan’s Island started out with a population of 7, we can find the doubling time by finding when the population is reach 14. By solving the equation

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graphically, we can find the value of t. We’ll do this by utilizing the Method of Intersection.

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Based on the graph, the population of Gilligan’s Island would double every 34.66 years since the doubling time is 34.66 years. This also means that the population would double again in an additional 34.66 years.

gilligan_13The population doubles from 14 to 28 in 69.32 – 34.66 or 34.66 years. For an increasing exponential function with a particular continuous rate, the doubling time is constant. Exponential functions with a different continuous rate also have doubling times. But those times are different from the one pictured above. Exponential functions with larger rates double faster and exponential functions with lower rates take longer to double.