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The number of people who have adopted a new fashion trend is increasing at a rate of r(t) people per month (where t is the time in months). What does int_(5)^(6)r(t)dt represent? Choose 1 answer: (A) The growth in the number of people to adopt the fashion trend during the sixth month B The time it took to change from 5 to 6 people who had adopted the fashion trend (C) The instantaneous rate of change of the number of people to adopt the fashion trend when t=6 months (D) The total number of people who have adopted the fashion trend by t=6 months

The number of people who have adopted a new fashion trend is increasing at a rate of r(t) r(t) people per month (where t t is the time in months).\newlineWhat does ∫56r(t)dt \int_{5}^{6} r(t) d t represent?\newlineChoose 11 answer:\newline(A) The growth in the number of people to adopt the fashion trend during the sixth month\newline(B) The time it took to change from 55 to 66 people who had adopted the fashion trend\newline(C) The instantaneous rate of change of the number of people to adopt the fashion trend when t=6 t=6 months\newline(D) The total number of people who have adopted the fashion trend by t=6 t=6 months

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Q. The number of people who have adopted a new fashion trend is increasing at a rate of r(t) r(t) people per month (where t t is the time in months).\newlineWhat does ∫56r(t)dt \int_{5}^{6} r(t) d t represent?\newlineChoose 11 answer:\newline(A) The growth in the number of people to adopt the fashion trend during the sixth month\newline(B) The time it took to change from 55 to 66 people who had adopted the fashion trend\newline(C) The instantaneous rate of change of the number of people to adopt the fashion trend when t=6 t=6 months\newline(D) The total number of people who have adopted the fashion trend by t=6 t=6 months
  1. Understand the integral: Understand the integral in the context of the problem. The integral of a rate of change function over an interval gives the total change over that interval. In this context, r(t)r(t) represents the rate of people adopting a new fashion trend per month. Therefore, the integral from 55 to 66 of r(t) dtr(t) \, dt represents the total number of people who have adopted the fashion trend from month 55 to month 66.
  2. Match interpretation to options: Match the interpretation of the integral to the given options.\newline(A) This option suggests that the integral represents the growth during the sixth month only, which is incorrect because the integral is over the interval from month 55 to month 66.\newline(B) This option is incorrect because the integral does not represent a time duration but rather a total change in the number of people.\newline(C) This option is incorrect because the integral does not represent an instantaneous rate of change but rather an accumulated total over an interval.\newline(D) This option is correct because the integral represents the total number of people who have adopted the fashion trend over the interval from month 55 to month 66.

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