P(t) = 260(1.04) ( 6 ⁄ 4 )t The function P models the population, in thousands, of a certain city t years after 2003. According to the model, the population is predicted to increase by 4% every n months. What is the value of n ?

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Accepted Answer

Choice A is correct. It’s given that the function P models the population, in thousands, of a certain city t years after 2003. The value of the base of the given exponential function, 1.04, corresponds to an increase of 4% for every increase of 1 in the exponent, ( 6 ⁄ 4 )t. If the exponent is equal to 0, then ( 6 ⁄ 4 )t = 0. Multiplying both sides of this equation by 4 ⁄ 6 yields t = 0. If the exponent is equal to 1, then ( 6 ⁄ 4 )t = 1. Multiplying both sides of this equation by 4 ⁄ 6 yields t = 4 ⁄ 6 , or t = 2 ⁄ 3 . Therefore, the population is predicted to increase by 4% every 2 ⁄ 3 of a year. It’s given that the population is predicted to increase by 4% every n months. Since there are 12 months in a year, 2 ⁄ 3 of a year is equivalent to ( 2 ⁄ 3 )(12), or 8, months. Therefore, the value of n is 8. Choice B is incorrect. This is the number of months in which the population is predicted to increase by 4% according to the model P(t) = 260(1.04) t , not P(t) = 260(1.04) ( 6 ⁄ 4 )t . Choice C is incorrect. This is the number of months in which the population is predicted to increase by 4% according to the model P(t) = 260(1.04) ( 4 ⁄ 6 )t , not P(t) = 260(1.04) ( 6 ⁄ 4 )t . Choice D is incorrect. This is the number of months in which the population is predicted to increase by 4% according to the model P(t) = 260(1.04) ( 1 ⁄ 6 )t , not P(t) = 260(1.04) ( 6 ⁄ 4 )t .

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