Question

A cup of coffee contains 100 milligrams of caffeine. Every approximately 6% of the amount of caffeine in the body is metabolized and eliminated
The rate of decay is ___%
The initial amount of caffeine in the body in mg and t represents the number of hours since a cup of coffee was consumed. Write C as a function of t
C(t) ___
Fill in the table
t 0 1 3 5
C(t) __ __ ___ __

Answer 1

To determine the rate of decay of caffeine in the body and write the function \( C(t) \), we can use the information provided:

1. The initial amount of caffeine is 100 mg.
2. 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:

1. For \( t = 0 \):
\[
C(0) = 100 \times (0.94)^0 = 100 \times 1 = 100 \, \text{mg}
\]

2. For \( t = 1 \):
\[
C(1) = 100 \times (0.94)^1 = 100 \times 0.94 = 94 \, \text{mg}
\]

3. For \( t = 3 \):
\[
C(3) = 100 \times (0.94)^3 \approx 100 \times 0.830584 = 83.06 \, \text{mg}
\]

4. 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 |