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Advanced Math / Nonlinear functions Difficulty: Hard

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 $y equals 0$ .

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

If during this time the population of the city decreases by a fixed percent each year, which of the following must be true?

A. $r is less than negative 1$

B. $negative 1 is less than r, which is less than 0$

C. $0 is less than r, which is less than 1$

D. $r is greater than 1$

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Explanation

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.