Let's begin by understanding the components of the exponential function:
[tex]\[ f(t) = 4300 \times (0.96)^{24t} \][/tex]
In this function:
- [tex]\( t \)[/tex] represents the time in days.
- 4300 is the initial quantity of whatever is being measured at the beginning ([tex]\( t = 0 \)[/tex]).
- 0.96 is the base of the exponent, which we'll focus on for this question.
- [tex]\( 24t \)[/tex] represents the number of time intervals (each day is split into 24 hours) over which the rate of change is applied. This means the rate of change per hour is considered.
Now, to understand what the 0.96 constant reveals about the rate of change:
The 0.96 is the decay factor of the quantity. This factor tells us that after each time interval, the quantity retains 96% of its previous amount, thus decreasing by 4% each interval. We can come to this conclusion because if the quantity were not changing, the factor would be 1 (100%).
To express the rate of change as a percentage, we subtract the decay factor from 1 and convert the result into a percentage:
Rate of Change (in %) = [tex]\((1 - 0.96) \times 100\)[/tex]
Rate of Change (in %) = [tex]\(0.04 \times 100\)[/tex]
Rate of Change (in %) = 4%
Therefore, the function is decreasing exponentially at a rate of 4% every hour. If we were looking at the daily rate of decrease, we would need to adjust the function to account for the 24-hour period explicitly, but the given function already includes this in the exponent.
