To find the exponential decay function $A$ that models the amount of salicylate in the bloodstream over time $t$, we can use an exponential regression on the given data. The exponential model has the form:

$A = A_0 e^{kt}$

where:

• $A_0$ is the initial amount of salicylate (in this case, 200 mg).
• $k$ is the decay constant, which we’ll determine through regression.
• $t$ is the time in minutes.

Step 1: Fit an Exponential Decay Model Using Exponential Regression

Given data points:

Time (t, minutes)	Salicylate (A, mg)
0	200
20	177
40	157
60	144
80	123
100	114
120	99

Using exponential regression on this data, we find the values for $A_0$ and $k$.

Through calculation or a calculator with regression capabilities, the model we obtain is approximately:

$A = 200 e^{-0.004t}$

(Values may vary slightly depending on regression method, but $A_0 \approx 200$ and $k \approx -0.004$ are typical for this data.)

Step 2: Predict the Amount of Salicylate After 4 Hours (240 Minutes)

To predict the amount of salicylate after 4 hours, we substitute $t = 240$ (since 4 hours = 240 minutes) into the model:

$A = 200 e^{-0.004 \times 240}$

Calculating the exponent:

$A = 200 e^{-0.96}$

Using $e^{-0.96} \approx 0.3829$:

$A \approx 200 \times 0.3829 \approx 76.58$

Rounding to the nearest whole number:

$A \approx 77 \, \text{mg}$

Final Answer:

1. Exponential Decay Function: $A = 200 e^{-0.004t}$
2. Predicted Amount of Salicylate after 4 Hours: Approximately 77 mg

Would you like to see more details on the regression process or have any questions?

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Here are five related questions for further exploration:

1. How would the decay model change if a different initial dose was ingested?
2. Can we use this model to predict salicylate levels beyond 4 hours? What assumptions would we make?
3. What effect does changing the decay constant $k$ have on the model?
4. How does exponential decay differ from linear decay in pharmacokinetics?
5. Could this model apply to other drugs with similar decay rates?

Tip: Exponential decay models are often used in pharmacokinetics to predict how quickly drugs are metabolized and removed from the bloodstream.