The function you provided,

$f(t) = 5200(0.9955)^{10t}$

describes a quantity that changes over time, $t$, measured in decades. The base $0.9955$ in this exponential function reveals information about the rate of change of the quantity.

Understanding the constant $0.9955$:

The number $0.9955$ represents a decay factor. In general, for an exponential function of the form:

$f(t) = A \cdot b^{kt}$

• $A$ is the initial amount,
• $b$ is the base (which affects the rate of growth or decay),
• $k$ is a constant (which adjusts the scale of change over time),
• $t$ is the time variable.

For $0.9955$, this base is slightly less than 1, indicating that the quantity decreases over time. In particular, $0.9955$ means that each time period (in this case, each decade, since the exponent is $10t$) the quantity is multiplied by $0.9955$, or 99.55% of its previous value.

Thus, the constant $0.9955$ shows that the quantity is experiencing a small, but consistent, decrease (about 0.45% decrease per decade).

Summary of the rate of change:

• The quantity decreases by approximately 0.45% each decade.
• The factor $0.9955$ reflects the decay rate, which is a reduction in the quantity over time.

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Would you like more details on exponential decay, or any related topics?

Here are some follow-up questions to explore:

1. What is the effect of changing the constant $0.9955$ to a lower value?
2. How can you calculate the total change in the quantity over multiple decades?
3. How does exponential decay differ from exponential growth?
4. What is the meaning of the constant $5200$ in this function?
5. How can we use logarithms to solve for $t$ when the quantity reaches a certain value?

Tip: In exponential decay problems, the rate of change is always the percentage decrease per unit time (here, per decade). Keep track of the base to understand how quickly the quantity decreases.