The question asks for situations that model exponential decay, which refers to a process where a quantity decreases at a rate proportional to its current value.

In the provided choices:

1. An investment that earns 5.99% annually compounded monthly - This represents exponential growth, not decay, since the investment is increasing in value.
2. The value of an automobile, which depreciates 15% each year - This is a case of exponential decay, as the value is decreasing by a fixed percentage annually.
3. A fish population that doubles every three months - This describes exponential growth, not decay, since the population is increasing.
4. The half-life of an element - This is a classic example of exponential decay, where the quantity of the element decreases by half over a consistent time interval.
5. The outbreak of an illness, which increases every two weeks - This describes exponential growth, as the number of infected individuals is increasing.
6. Algae in a pond, which increases over a month - This also describes exponential growth, as the algae population is increasing.

Thus, the correct choices that model exponential decay are:

• The value of an automobile, which depreciates 15% each year
• The half-life of an element

Would you like further explanation on any part of this? Here are 5 related questions to expand on this topic:

1. What is the formula for exponential decay and how is it applied?
2. How does exponential decay differ from linear decay?
3. Can you provide more real-world examples of exponential decay?
4. How does the rate of decay affect the time it takes for a substance to reach half its initial value?
5. How is exponential growth different from exponential decay in terms of mathematical modeling?

Tip: Exponential decay can be modeled by the equation $N(t) = N_0 e^{-kt}$, where $N_0$ is the initial amount, $k$ is the decay constant, and $t$ is time.