The population P of a certain city y years after the last census is modeled by the equation below, where

Question

The population P of a certain city y years after the last census is modeled by the equation below, where r is a constant and $P subscript 0$ is the population when $y equals 0$ .

$P equals, P subscript 0, times, open parenthesis, 1 plus r, close parenthesis, to the power y$

Accepted Answer

Choice B is correct. The term (1 + r ) represents a percent change. Since the population is decreasing, the percent change must be between 0% and 100%. When the percent change is expressed as a decimal rather than as a percent, the percentage change must be between 0 and 1. Because (1 + r ) represents percent change, this can be expressed as 0 < 1 + r < 1. Subtracting 1 from all three terms of this compound inequality results in -1 < r < 0. Choice A is incorrect. If r < -1, then after 1 year, the population P would be a negative value, which is not possible. Choices C and D are incorrect. For any value of r > 0, 1 + r > 1, and the exponential function models growth for positive values of the exponent. This contradicts the given information that the population is decreasing.

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