Let's go through the problem step by step.
Part A: The rate of decay is ___%
The rate of decay is given as 6%. Therefore:
Part A: The rate of decay is 6%
Part B: The initial amount of caffeine consumed in this scenario is ___ mg.
According to the problem, a typical cup of coffee contains about 100 mg of caffeine.
Part B: The initial amount of caffeine consumed in this scenario is 100 mg.
Part C: Write C as a function of t.
To model the caffeine decay, we use the formula for exponential decay:
\[ C(t) = C_0 \cdot (1 - r)^t \]
where:
- \( C_0 \) is the initial amount of caffeine (100 mg),
- \( r \) is the decay rate (0.06),
- \( t \) is the time in hours.
Putting these values into the formula:
\[ C(t) = 100 \cdot (1 - 0.06)^t \] \[ C(t) = 100 \cdot (0.94)^t \]
Part C: C(t) = 100 * (0.94)^t
Part D: Fill out the table below. Round your answer to the nearest hundredth.
Using the equation we derived, we will now calculate \( C(t) \) for \( t = 0, 1, 3, 5 \):
-
For \( t = 0 \): \[ C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100.00 \]
-
For \( t = 1 \): \[ C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \]
-
For \( t = 3 \): \[ C(3) = 100 \cdot (0.94)^3 \approx 100 \cdot 0.830584 = 83.06 \]
-
For \( t = 5 \): \[ C(5) = 100 \cdot (0.94)^5 \approx 100 \cdot 0.735091 = 73.51 \]
Now, we can fill in the table:
| t | 0 | 1 | 3 | 5 | |---|-------|-------|-------|-------| | C(t) | 100.00 | 94.00 | 83.06 | 73.51 |
So, the completed table is:
t: 0, 1, 3, 5
C(t): 100.00, 94.00, 83.06, 73.51