The function given is:

$f(t) = 2100 \left(0.25\right)^{\frac{t}{12}}$

This represents a quantity that changes over time, where:

• $f(t)$ is the value of the quantity after $t$ months.
• 2100 is the initial value of the quantity when $t = 0$.
• The base $0.25$ indicates the rate at which the quantity is changing over time.
• The exponent $\frac{t}{12}$ suggests that the change occurs with respect to months, and it is based on the fraction of the year (since there are 12 months in a year).

Interpretation of $0.25$:

The constant $0.25$ reveals that the quantity is changing at a rate where each year (12 months), the quantity is multiplied by $0.25$. In other words:

• After 12 months (1 year), the quantity will be $2100 \times 0.25 = 525$.
• After another 12 months (2 years), the quantity will be $2100 \times (0.25)^2 = 131.25$, and so on.

This indicates that the quantity is decreasing by a factor of $0.25$ each year. Hence, the quantity is decaying or shrinking over time at a rate of 75% per year, because it retains only 25% of its value after each year.

To summarize:

• The constant $0.25$ reveals that the quantity is decreasing at a rate of 75% per year (since $1 - 0.25 = 0.75$).

Would you like more details about how this function behaves over time? Here are some additional questions:

1. How would the graph of this function look?
2. What happens if the base is greater than 1 instead of 0.25?
3. How can we generalize this type of exponential decay function?
4. How does the initial value (2100) influence the function?
5. What would happen to the quantity if the exponent had a different factor than $\frac{t}{12}$?

Tip: Exponential decay functions are typically used to model situations like population decline, radioactive decay, or the depreciation of an asset.