Consider the following problem: The population of ants in Chloe's ant farm changes at a rate of r(t)=-15.8*0.9^(t) ants per month (where t is time in months). At time t=0, the ant farm's population is 150 ants. How many ants are in the farm at t=4 ? Which expression can we use to solve the problem? Choose 1 answer: (A) int_(0)^(4)r(t)dt (B) int_(3)^(4)r(t)dt (C) 150+int_(3)^(4)r(t)dt (D) 150+int_(0)^(4)r(t)dt

Question

Consider the following problem: The population of ants in Chloe's ant farm changes at a rate of  r(t)=-15.8*0.9^(t) ants per month (where  t is time in months). At time  t=0, the ant farm's population is 150 ants. How many ants are in the farm at  t=4 ? Which expression can we use to solve the problem? Choose 1 answer: (A)  int_(0)^(4)r(t)dt (B)  int_(3)^(4)r(t)dt (C)  150+int_(3)^(4)r(t)dt (D)  150+int_(0)^(4)r(t)dt

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Initial Population and Rate Function: To find the number of ants at t=4, we need to account for the initial population and the change in population over time. The initial population is given as 150 ants at t=0. The rate of change of the population is given by the function r(t)=-15.8*0.9^(t). To find the total change in population from t=0 to t=4, we need to integrate the rate function over this interval.Definite Integral Calculation: We will use the definite integral to calculate the total change in population from t=0 to t=4. The integral of the rate function r(t) over the interval from 0 to 4 will give us the net change in the population during this time period.Total Number of Ants at t=4: The correct expression to calculate the total number of ants at t=4 is the initial population plus the integral of the rate function from t=0 to t=4. Mathematically, this is represented as: 150 + ∫_(0)^(4)r(t)dtCorrect Choice Explanation: Looking at the given options, we can see that option (D) matches the expression we derived: (D) 150 + ∫_(0)^(4)r(t)dt This is the correct choice because it starts with the initial population and adds the change in population from t=0 to t=4.Incorrect Options Explanation: To confirm, options (A), (B), and (C) are incorrect because: (A) does not include the initial population. (B) starts integrating from t=3, which would not account for the change from t=0 to t=3. (C) includes the initial population but starts integrating from t=3, which also does not account for the change from t=0 to t=3.

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