Final answer:
The given exponential function y = 400(0.37)^x represents exponential decay with a percentage rate of decrease of 63%. A graph of this function would show a decreasing curve. The 'rule of 70' can be used to estimate doubling times for quantities growing exponentially.
Step-by-step explanation:
The exponential function given is y = 400(0.37)^x. In this function, the base of the exponent is 0.37, which is less than 1. This indicates an exponential decay rather than exponential growth, because the value of y decreases as x increases. To find the percentage rate of decrease, subtract the base from 1 and then multiply by 100. Therefore, the percentage rate of decrease is (1 - 0.37) × 100 = 63%.
To illustrate exponential decay, we would draw a graph with the y-axis representing the value of y and the x-axis representing the time or growth period (x). The graph would show a curve that starts high when x is small and then falls off rapidly as x increases. As for the specific request to shade the area representing P(x < 0.40) in a probability context, this would typically require a different kind of function related to probability distribution, which is not provided here. Understanding the concept of exponential functions is crucial for evaluating growth or decline over time. For example, if a quantity grows by a fixed percentage every time period, that is exponential growth. If it decreases by a fixed percentage, it's exponential decay. The rule of 70 is often used to estimate the doubling time of a quantity growing exponentially.