### Population Change Study in OhioA small town in Ohio commissioned an actuarial firm to conduct a study that modeled the rate of change of the town’s population. The study found that the town’s population (measured in thousands of people) can be modeled by the function:\[ P(t) = -\frac{1}{3}t^3 + 64t + 3000, \]where \( t \) is measured in years.#### Tasks:**a.** Find the rate of change function \(P'(t)\) of the population function.**b.** Find \( P'(1) \), \( P'(2) \), \( P'(3) \), and \( P'(4) \). Interpret what the results mean for the town.**c.** Find \( P''(1) \), \( P''(2) \), \( P''(3) \), and \( P''(4) \). Interpret what the results mean for the town’s population.

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15. A small town in Ohio commissioned an actuarial firm to conduct a study that modeled the rate of change of the town's popula tion. The study found that the town's population (measured in thousands of people) can be modeled by the function P(t) = ³ +64t+3000, where t is measured in years. a. Find the rate of change function P'(t) of the population function. b. Find P'(1), P'(2), P'(3), and P'(4). Interpret what the results mean for the town. c. Find P" (1), P" (2) P" (3), and P" (4). Interpret what the results mean for the town's population.