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The number of people exposed to a certain illness is increasing at a rate of r(t) people per day (where t is the time in days). At t=12, there were 115 people exposed to the illness. What does 115+int_(12)^(20)r(t)dt represent? Choose 1 answer: (A) The change in the number of people exposed to the illness between days 12 and 20 B The change in the rate at which people are exposed to the illness between days 12 and 20 (C) The time it takes for 20 more people to be exposed to the illness (D) The number of people exposed to the illness at t=20

The number of people exposed to a certain illness is increasing at a rate of r(t) r(t) people per day (where t t is the time in days). At t=12 t=12 , there were 115115 people exposed to the illness.\newlineWhat does 115+∫1220r(t)dt 115+\int_{12}^{20} r(t) d t represent?\newlineChoose 11 answer:\newline(A) The change in the number of people exposed to the illness between days 1212 and 2020\newline(B) The change in the rate at which people are exposed to the illness between days 1212 and 2020\newline(C) The time it takes for 2020 more people to be exposed to the illness\newline(D) The number of people exposed to the illness at t=20 t=20

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Q. The number of people exposed to a certain illness is increasing at a rate of r(t) r(t) people per day (where t t is the time in days). At t=12 t=12 , there were 115115 people exposed to the illness.\newlineWhat does 115+∫1220r(t)dt 115+\int_{12}^{20} r(t) d t represent?\newlineChoose 11 answer:\newline(A) The change in the number of people exposed to the illness between days 1212 and 2020\newline(B) The change in the rate at which people are exposed to the illness between days 1212 and 2020\newline(C) The time it takes for 2020 more people to be exposed to the illness\newline(D) The number of people exposed to the illness at t=20 t=20
  1. Understand Expression: First, let's understand the expression 115+∫1220r(t)dt115+\int_{12}^{20}r(t)dt. The integral ∫1220r(t)dt\int_{12}^{20}r(t)dt represents the total change in the number of people exposed to the illness from day 1212 to day 2020. The number 115115 represents the number of people already exposed to the illness at t=12t=12. Therefore, when we add 115115 to the integral, we are adding the initial number of exposed people to the total change in the number of exposed people from day 1212 to day 2020.
  2. Analyze Answer Choices: Now, let's analyze the answer choices given:\newline(A) The change in the number of people exposed to the illness between days 1212 and 2020\newline(B) The change in the rate at which people are exposed to the illness between days 1212 and 2020\newline(C) The time it takes for 2020 more people to be exposed to the illness\newline(D) The number of people exposed to the illness at t=20t=20\newlineThe integral alone (∫1220r(t)dt)(\int_{12}^{20}r(t)dt) would represent the change in the number of people exposed to the illness between days 1212 and 2020, which corresponds to option (A). However, since we are adding 115115 to the integral, we are not looking for just the change, but the total number of people exposed by day 2020. This means that option (A) is not the correct interpretation of the expression.
  3. Option (A) Explanation: Option (B) is incorrect because the integral represents the accumulation of the rate of change, not the change in the rate itself. The rate of change is r(t)r(t), and the integral of r(t)r(t) gives us the total change over the interval, not the change in the rate.
  4. Option (B) Explanation: Option (C) is incorrect because the expression does not represent a time duration but rather a number of people. The integral does not give us the time it takes for a certain number of people to be exposed; it gives us the number of people exposed over a certain time interval.
  5. Option (C) Explanation: Option (D) is the correct interpretation of the expression. The integral ∫1220r(t) dt\int_{12}^{20} r(t) \, dt gives us the total number of people exposed to the illness from day 1212 to day 2020, and when we add the 115115 people who were already exposed by day 1212, we get the total number of people exposed to the illness by day 2020.

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