Final answer:
To find the instantaneous rate of change of the population in the year 2025, we need to find the derivative of the population function with respect to time. The derivative of the population function is found using the chain rule, and then we substitute the value of t = 12 into the derivative expression to find the instantaneous rate of change. The instantaneous rate of change in the year 2025 is approximately 3.58.
Step-by-step explanation:
To find the instantaneous rate of change of the population in the year 2025, we need to find the derivative of the population function with respect to time. The population function is given as p(t) = 317e^0.01t. To find the derivative, we use the chain rule, which states that if we have a function f(g(t)), then the derivative of f(g(t)) is f'(g(t)) * g'(t). In this case, f(t) = 317e^t, so the derivative is f'(t) = 317 * e^t, and g(t) = 0.01t, so the derivative is g'(t) = 0.01. Therefore, the derivative of p(t) is p'(t) = f'(g(t)) * g'(t) = 317 * e^(0.01t) * 0.01. To find the instantaneous rate of change in the year 2025, we substitute t = 12 (2025 - 2013) into p'(t) and evaluate the expression.
p'(12) = 317 * e^(0.01 * 12) * 0.01 = 317 * e^0.12 * 0.01 = 3.17 * e^0.12
Using a calculator, we find that e^0.12 is approximately 1.127. So the instantaneous rate of change in the year 2025 is 3.17 * 1.127 = 3.58059. Rounded to two decimal places, the instantaneous rate of change is approximately 3.58.