In November 2010 , a "tiger summit" was held in St....

In November 2010 , a "tiger summit" was held in St. Petersburg, Russia. "In $1900,$ there were 100,000 wild tigers worldwide; in 2010 the number was 3200 . (a) Assuming the tiger population has decreased exponentially, find a formula for $f(t),$ the number of wild tigers $t$ years since $1900 .$ (b) Between 2000 and 2010 , the number of wild tigers decreased by $40 \%$. Is this percentage larger or smaller than the decrease in the tiger population predicted by your answer to part (a)?

In November 2010 , a "tiger summit" was held in St. Petersburg, Russia. "In $1900,$ there were 100,000 wild tigers worldwide; in 2010 the number was 3200 .
(a) Assuming the tiger population has decreased exponentially, find a formula for $f(t),$ the number of wild tigers $t$ years since $1900 .$
(b) Between 2000 and 2010 , the number of wild tigers decreased by $40 \%$. Is this percentage larger or smaller than the decrease in the tiger population predicted by your answer to part (a)?

Key Concepts

Model Validation

Model validation is the process of comparing the predictions of a mathematical model with real-world data. It involves checking if the percentage decrease or other trends predicted by an exponential decay formula accurately reflect the observed changes. This step is crucial in determining the reliability and accuracy of the model in representing the underlying phenomenon.

Percentage Change

The concept of percentage change is central to understanding how a quantity evolves over time. In exponential decay problems, it is used to compare the proportionate decrease over a specific period with the changes predicted by the model. This involves calculating the ratio of the quantity at two different times and expressing the decrease as a percentage of the earlier value.

Parameter Estimation

Parameter estimation involves determining the constants in a mathematical model using known data points. In the context of exponential decay, this typically means using initial conditions and subsequent measurements to solve for the decay constant r or the base of the exponential function, thus accurately fitting the model to the observed data.

Exponential Decay

Exponential decay describes a process in which the rate of decrease of a quantity is proportional to the current amount of that quantity. In population modeling, this means that populations decrease by a fixed percentage over equal time intervals, leading to a formula of the form f(t) = f(0) · e^(rt), where r is a negative constant representing the decay rate.

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