To determine the rate of decay in the exponential function $A(t) = P(0.82)^t$, we need to look at the base of the exponent. The general form for exponential decay is:

$A(t) = P \cdot e^{rt}$

Where:

• $A(t)$ is the amount at time $t$,
• $P$ is the principal (initial amount),
• $r$ is the rate of decay,
• $e$ is the base of the natural logarithm (when the function is written in terms of natural exponentials).

In the given equation, $A(t) = P(0.82)^t$, the base of the exponential is 0.82, indicating that the quantity is decreasing by a factor of 0.82 each time. This means the amount is decreasing by 18% each period.

To confirm this, subtract the base from 1:

$1 - 0.82 = 0.18$

So, the rate of decay is 18%.

Correct answer: 18%

---

Would you like more details on exponential decay or compounding functions? Here are some questions related to this topic:

1. How would you modify the equation to model exponential growth instead of decay?
2. How do you calculate the amount after a certain number of periods using the exponential decay formula?
3. What is the difference between exponential decay and linear decay?
4. How can you derive the rate of decay if you have two data points from an exponential decay function?
5. What is the effect of changing the base from 0.82 to a smaller value on the decay rate?

Tip: When analyzing exponential functions, always identify the base of the exponent to determine whether it is a decay (if the base is less than 1) or growth (if the base is greater than 1).