The function \( P(t)=\frac{25(5 t+8)}{2 t+1} \) models the population, in thousands, of a town \( t \) years since 1990 . a) Find the initial population. b) Find the average rate of change of the population from 1995 to 2005 . c) Find the rate at which the population is changing in the year 2000 .

To answer your questions, let's solve step by step: a) The initial population corresponds to \( t = 0 \) (which is the year 1990). We substitute \( t = 0 \) into the function \( P(t) \): \[ P(0) = \frac{25(5(0)+8)}{2(0)+1} = \frac{25(8)}{1} = 200. \] So, the initial population in 1990 is 200 thousand, or 200,000 people. b) To find the average rate of change of the population from 1995 to 2005, we first calculate the populations at \( t = 5 \) (1995) and \( t = 15 \) (2005): \[ P(5) = \frac{25(5(5)+8)}{2(5)+1} = \frac{25(25+8)}{10+1} = \frac{25 \times 33}{11} = 75. \] \[ P(15) = \frac{25(5(15)+8)}{2(15)+1} = \frac{25(75+8)}{30+1} = \frac{25 \times 83}{31} = 66.94 \text{ (approximately)}. \] Now we compute the average rate of change: \[ \text{Average Rate of Change} = \frac{P(15) - P(5)}{15 - 5} = \frac{66.94 - 75}{10} = -0.606 \text{ thousand people per year (approximately)}. \] c) To find the rate of change of the population in the year 2000, we calculate the derivative \( P'(t) \) and evaluate it at \( t = 10 \): First, we apply the quotient rule: \[ P(t) = \frac{u(t)}{v(t)} = \frac{25(5t + 8)}{2t + 1}, \quad u(t) = 25(5t + 8), \quad v(t) = 2t + 1. \] Using the quotient rule \( P'(t) = \frac{u'v - uv'}{v^2} \): \( u' = 125, \quad v' = 2 \). So, \[ P'(t) = \frac{(125)(2t + 1) - (25(5t + 8))(2)}{(2t + 1)^2}. \] Plugging in \( t = 10 \): \[ P'(10) = \frac{125(21) - (25(58))(2)}{(21)^2} = \frac{2625 - 2900}{441} = \frac{-275}{441}. \] Therefore, the rate of change in 2000 is approximately -0.623 thousand people per year. Now, if you'd like some extra insights: Fun Fact: The methodology we used for calculating the average rate of change is foundational in calculus, especially in understanding how populations grow or shrink over time. It's a great example of how math reflects real-world scenarios! Real-World Link: The function you’re working with is similar to those used by urban planners and demographers, as they model city growth to help allocate resources, plan infrastructure, and estimate future needs based on historical data trends! Who knew math could inform our future city designs?