what exponential decay function fits the following measurement points (time, weight): (0, 5960) (1, 2440) (70, 1390)

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.

What exponential decay function fits the following measurement points (time, weight): (0, 5960), (1, 2440), (70, 1390)?

Explore how to fit an exponential decay function for the measurement points (0, 5960), (1, 2440), and (70, 1390). This problem introduces the concept of exponential decay, involves logarithmic calculations, and applies the decay constant to determine the best fit.

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