Let's go through each part of the question step by step.
Part A
The rate of decay is the percentage of the substance that is lost per hour. According to the problem, 6% of caffeine is metabolized and eliminated every hour. Therefore, the rate of decay is:
Rate of Decay = 6%
Part B
The initial amount of caffeine consumed is stated in the problem. A typical cup of coffee contains about 100 milligrams (mg) of caffeine.
Initial amount of caffeine = 100 mg
Part C
We need to express the amount of caffeine \( C \) as a function of time \( t \) in hours. Since 6% is eliminated every hour, 94% remains.
The decay function can be expressed as: \[ C(t) = C_0 \cdot (1 - r)^t \] Where:
- \( C_0 \) is the initial amount of caffeine (100 mg),
- \( r \) is the decay rate in decimal form (6% is 0.06),
- \( t \) is time in hours.
So, the function is: \[ C(t) = 100 \cdot (0.94)^t \]
Part D
Now, we will calculate \( C(t) \) for the specified values of \( t \): 0, 1, 3, and 6.
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For \( t = 0 \): \[ C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100 \]
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For \( t = 1 \): \[ C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94 \]
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For \( t = 3 \): \[ C(3) = 100 \cdot (0.94)^3 \approx 100 \cdot 0.830584 = 83.06 \]
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For \( t = 6 \): \[ C(6) = 100 \cdot (0.94)^6 \approx 100 \cdot 0.528987 = 52.90 \]
Here’s the completed table:
| t | 0 | 1 | 3 | 6 | |----|-----|-----|------|------| | C(t) | 100 | 94 | 83.06 | 52.90 |
Feel free to ask if you have any other questions!
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