Step 1: Formulate the exponential model

The general form of the exponential model is: $C = a \cdot b^t$ Where:

• $C$ is the amount of caffeine (mg),
• $a$ is the initial amount of caffeine,
• $b$ is the decay factor,
• $t$ is the time elapsed (hours).

From the table:

• At $t = 0$, $C = 150$. Therefore, $a = 150$.
• The value of $b$ can be determined using another point from the table.

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Step 2: Solve for $b$

Using the point at $t = 1$, $C = 132$: $132 = 150 \cdot b^1$ $b = \frac{132}{150} = 0.88$

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Step 3: Write the model equation

Substitute $a = 150$ and $b = 0.88$ into the equation: $C = 150 \cdot 0.88^t$

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Step 4: Analyze the impact of drinking two cups of coffee

If Forest drinks two cups of coffee, the initial amount of caffeine ($a$) will double because each cup contains 150 mg of caffeine. Thus, the new $a$ would be $2 \cdot 150 = 300$. However, the decay factor ($b$) remains the same, as the rate at which caffeine is metabolized does not depend on the initial quantity.

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Final Answer:

1. The equation of the exponential function is: $C = 150 \cdot 0.88^t$

2. If Forest drinks two cups of coffee, the initial amount of caffeine ($a$) will change, but the decay factor ($b$) will remain the same.

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Additional Questions:

1. How does the half-life of caffeine relate to the decay factor $b$?
2. Can you find the half-life of caffeine based on the given data?
3. What happens to the graph of the function if Forest drinks three cups of coffee instead of one?
4. How would you interpret $b = 0.88$ in terms of caffeine metabolism?
5. What other real-world processes can be modeled using an exponential decay function?

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Tip:

To better understand exponential decay, compare the caffeine amounts after each hour and see how they form a consistent percentage decrease. This is the hallmark of exponential decay.