The approval rating of a certain politician is changing at a rate of r(t) percent of the population per month (where t is the time in months). What does int_(0)^(3)r(t)dt represent? Choose 1 answer: (A) The rate of change of the approval rating of the politician when t=3 months (B) The average rate of change of the approval rating during the first three months (C) The percent of the population who approves of the politician when t=3 months (D) The change in the percent of the population who approves of the politician during the first three months

Question

The approval rating of a certain politician is changing at a rate of  r(t) percent of the population per month (where  t is the time in months). What does  int_(0)^(3)r(t)dt represent? Choose 1 answer: (A) The rate of change of the approval rating of the politician when  t=3 months (B) The average rate of change of the approval rating during the first three months (C) The percent of the population who approves of the politician when  t=3 months (D) The change in the percent of the population who approves of the politician during the first three months

Bytelearn · Accepted Answer

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 net change over that interval. In this case, r(t) represents the rate of change of the politician's approval rating over time, so the integral from 0 to 3 of r(t) dt represents the total change in approval rating over the first three months.Match with answer choices: Match the integral's meaning to the given answer choices. The integral does not give us the rate of change at a specific time (which would be r(t) at t=3), so option (A) is incorrect. It also does not give us the average rate of change, which would be the integral divided by the time interval, so option (B) is incorrect. It does not give us the percent of the population who approves of the politician at a specific time, so option (C) is incorrect. The integral does represent the total change in approval rating over the time interval, which matches option (D).Confirm correct choice: Confirm the correct answer choice. Since the integral of a rate of change function gives the net change, and we are looking at the change from time 0 to time 3, the correct answer is (D) The change in the percent of the population who approves of the politician during the first three months.

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