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9-81.

When rabbits were first brought to Australia, they had no natural enemies. There were about $80,000$ rabbits in 1866. Two years later, in 1868, the population had grown to over $2,400,000$! Homework Help ✎

1. Why is an exponential equation a better model for this situation than a linear one? Is a sine function better or worse? Why?

A sine function shows when a population rises and falls.

2. Write an exponential equation for the number of rabbits $t$ years after 1866.

Recall the general exponential equation:
$y = ab^{x} + c$

Substitute the initial number of rabbits and a point you know will be on the graph in the proper places, and solve for $b$.

3. How many rabbits does your model predict were present in 1871?

$≈ 394$ million

4. According to your model, in what year was the first pair of rabbits introduced into Australia? Is this reasonable?

Solve your exponential equation for $t$ using $2$ as the number of rabbits.

5. Actually, $24$ rabbits were introduced in 1859, so your model is not perfect, but it is close. Is your model useful for predicting how many rabbits there are now? Explain.

Will the population of rabbits continue growing infinitely as the exponential equation suggests?