An exponential decay function is generally of the form:

$y(t) = y_0 \cdot e^{-kt}$

where:

• $y(t)$ is the quantity at time $t$,
• $y_0$ is the initial quantity at $t = 0$,
• $k$ is the decay constant (rate),
• $t$ is the time.

Given the points:

• $(0, 5960)$,
• $(1, 2440)$,
• $(70, 1390)$,

we can solve for $y_0$ and $k$.

Step 1: Find $y_0$

From $(0, 5960)$, when $t = 0$: $y(0) = y_0 = 5960$

So, the equation becomes: $y(t) = 5960 \cdot e^{-kt}$

Step 2: Solve for $k$

Using the second point $(1, 2440)$, substitute into the equation:

$2440 = 5960 \cdot e^{-k \cdot 1}$

Solving for $k$: $\frac{2440}{5960} = e^{-k}$

$0.409395 = e^{-k}$

Taking the natural logarithm of both sides: $\ln(0.409395) = -k$

$k = -\ln(0.409395) \approx 0.892$

Step 3: Verify with the third point

Now that we have $k$, we can verify the fit with the third point $(70, 1390)$.

The function is: $y(t) = 5960 \cdot e^{-0.892t}$

Substituting $t = 70$: $y(70) = 5960 \cdot e^{-0.892 \cdot 70}$

Evaluating this will confirm the validity of the fit. Let's compute that next.When using $k = 0.892$, the calculation yields an extremely small value for $y(70)$, approximately $4.55 \times 10^{-24}$, which is clearly much smaller than the observed value of 1390. This indicates that $k = 0.892$ is too large, suggesting a slower decay rate than initially calculated.

Let me try recalculating $k$ using both the first two points while considering the third as a constraint. I'll compute it more carefully.After recalculating using all three points, the fitted values are:

• $y_0 = 5960$ (which matches the initial condition),
• $k \approx 0.893$, similar to the earlier estimate.

The decay rate $k$ is slightly more refined, but this still results in an extremely fast decay when extrapolated to $t = 70$. Given this, it's possible that the model might not perfectly follow a single exponential decay across the whole range, or some data points might reflect non-exponential behavior.

Would you like further clarification, or to adjust the model for a better fit?

Relative Questions:

1. How does the decay constant $k$ affect the rate of decrease in an exponential decay function?
2. What assumptions are involved in modeling with an exponential decay function?
3. Could a different mathematical model better fit the data points?
4. How can the exponential decay equation be used in real-world scenarios (e.g., radioactive decay)?
5. What happens when $k$ becomes very small or very large?

Tip: If the model doesn't fit well, consider a multi-phase decay model or using logarithmic scaling for better visualization.