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 is the population when
.
If during this time the population of the city decreases by a fixed percent each year, which of the following must be true?
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A
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B
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C
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D
Correct answer: B
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.
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