To determine the rate of decay of caffeine in the body and write the function \( C(t) \), we can use the information provided:
- The initial amount of caffeine is 100 mg.
- Approximately 6% of the caffeine is metabolized and eliminated every hour.
From this information, we can find the rate of decay \( r \):
- Since 6% is metabolized, 94% remains after one hour. This gives a decay rate of 94%.
To express this as a decimal, we have: \[ C(t) = C(0) \times (0.94)^t \] Where \( C(0) = 100 , \text{mg} \).
Thus, the function \( C(t) \) can be written as: \[ C(t) = 100 \times (0.94)^t \]
Now we will calculate \( C(t) \) for the specific values of \( t \) given in the table:
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For \( t = 0 \): \[ C(0) = 100 \times (0.94)^0 = 100 \times 1 = 100 , \text{mg} \]
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For \( t = 1 \): \[ C(1) = 100 \times (0.94)^1 = 100 \times 0.94 = 94 , \text{mg} \]
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For \( t = 3 \): \[ C(3) = 100 \times (0.94)^3 \approx 100 \times 0.830584 = 83.06 , \text{mg} \]
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For \( t = 5 \): \[ C(5) = 100 \times (0.94)^5 \approx 100 \times 0.735091 = 73.51 , \text{mg} \]
Filling in the table:
| t | 0 | 1 | 3 | 5 | |----|-----|------|--------|-------| | C(t) | 100 | 94 | 83.06 | 73.51 |
Thus, the completed table is as follows:
| t | 0 | 1 | 3 | 5 | |----|-----|-----|--------|-------| | C(t) | 100 | 94 | 83.06 | 73.51 |