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 instantaneous rate of change of the number of people to adopt the fashion trend when t=6 months (B) The total number of people who have adopted the fashion trend by t=6 months (C) The growth in the number of people to adopt the fashion trend during the sixth month (D) The time it took to change from 5 to 6 people who had adopted the fashion trend

Question

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 instantaneous rate of change of the number of people to adopt the fashion trend when  t=6 months (B) The total number of people who have adopted the fashion trend by  t=6 months (C) The growth in the number of people to adopt the fashion trend during the sixth month (D) The time it took to change from 5 to 6 people who had adopted the fashion trend

Bytelearn · Accepted Answer

Understand Problem Context: Understand the meaning of 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) represents the rate of change of the number of people adopting a new fashion trend per month. Therefore, the integral from 5 to 6 of r(t) dt represents the total change in the number of people who have adopted the fashion trend from month 5 to month 6.Match Meaning to Choices: Match the meaning of the integral to the given answer choices. (A) The instantaneous rate of change of the number of people to adopt the fashion trend when t=6 months - This choice describes a derivative at a specific point in time, not an integral. (B) The total number of people who have adopted the fashion trend by t=6 months - This choice is incorrect because the integral does not give the total number by a certain time, but the change over an interval of time. (C) The growth in the number of people to adopt the fashion trend during the sixth month - This choice is incorrect because the integral is over the interval from month 5 to month 6, not just during the sixth month. (D) The time it took to change from 5 to 6 people who had adopted the fashion trend - This choice is incorrect because the integral measures the number of people, not the time it takes for a change to occur.

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