Consider the following problem: The population of a town grows at a rate of r(t)=t^(2)+2t people per year (where t is time in years). At time t=3, the town's population is 2300 people. By how much did the population grow between years 3 and 10 ? Which expression can we use to solve the problem? Choose 1 answer: (A) r(10)-r(3) (B) int_(3)^(10)r(t)dt (C) int r(t)dt (D) 2300+int_(3)^(10)r(t)dt

Question

Consider the following problem: The population of a town grows at a rate of  r(t)=t^(2)+2t people per year (where  t is time in years). At time  t=3, the town's population is 2300 people. By how much did the population grow between years 3 and 10 ? Which expression can we use to solve the problem? Choose 1 answer: (A)  r(10)-r(3) (B)  int_(3)^(10)r(t)dt (C)  int r(t)dt (D)  2300+int_(3)^(10)r(t)dt

Bytelearn · Accepted Answer

Calculate total growth: To find the growth of the population between years 3 and 10, we need to calculate the total increase over that period. The rate of growth is given by the function r(t) = t^2 + 2t. We need to integrate this rate of growth from t = 3 to t = 10 to find the total increase in population.Integral of growth rate: The correct expression to calculate the total growth of the population from year 3 to year 10 is the integral of the rate of growth function r(t) from t = 3 to t = 10. This is represented by the integral expression ∫ from 3 to 10 of r(t) dt.Antiderivative of r(t): The integral of r(t) = t^2 + 2t from t = 3 to t = 10 will give us the total number of people added to the population during this time. We can calculate this integral using the fundamental theorem of calculus.Evaluate upper limit: First, we find the antiderivative of r(t). The antiderivative of t^2 is (1/3)t^3, and the antiderivative of 2t is t^2. So the antiderivative of r(t) is (1/3)t^3 + t^2.Evaluate lower limit: Next, we evaluate the antiderivative at the upper limit of integration, which is t = 10. Plugging in 10, we get (1/3)(10)^3 + (10)^2 = (1/3)(1000) + 100 = 333.33 + 100 = 433.33.Subtract values: Then, we evaluate the antiderivative at the lower limit of integration, which is t = 3. Plugging in 3, we get (1/3)(3)^3 + (3)^2 = (1/3)(27) + 9 = 9 + 9 = 18.Final population growth: Now, we subtract the value of the antiderivative at t = 3 from the value at t = 10 to find the total population growth between these years. 433.33 - 18 = 415.33.Correct expression: Therefore, the population grew by 415.33 people from year 3 to year 10. The correct expression to use for solving this problem is (B) ∫ from 3 to 10 of r(t) dt.

Full solution

More problems from Solve one-step multiplication and division equations: word problems

Related topics