Question
Part D: Fill out the table below. Round your answer to the nearest hundredth.
t
0 1 3 5
C(t)
t
0 1 3 5
C(t)
Answers
GPT-4o mini
It seems you are looking for values of \( C(t) \) for specific values of \( t \). However, you haven't provided the formula or function for \( C(t) \).
If you can provide the function or any additional context needed to calculate \( C(t) \), I would be glad to help fill out the table!
If you can provide the function or any additional context needed to calculate \( C(t) \), I would be glad to help fill out the table!
A typical cup of coffee contains about 100 milligrams (mg) of caffeine.
Every hour, approximately 6%
of the amount of caffeine in the body is metabolized and eliminated.
Part A: The rate of decay is
%
Part B: The initial amount of caffeine consumed in this scenario is
mg.
Part C: Let C
represent the 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) =
Part D: Fill out the table below. Round your answer to the nearest hundredth.
t
0 1 3 5
C(t)
Every hour, approximately 6%
of the amount of caffeine in the body is metabolized and eliminated.
Part A: The rate of decay is
%
Part B: The initial amount of caffeine consumed in this scenario is
mg.
Part C: Let C
represent the 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) =
Part D: Fill out the table below. Round your answer to the nearest hundredth.
t
0 1 3 5
C(t)
GPT-4o mini
Let's go through the parts step-by-step.
### Part A: The rate of decay
The problem states that every hour, approximately 6% of the amount of caffeine in the body is metabolized and eliminated. In terms of the decay rate:
- **Rate of Decay**: 6%
### Part B: The initial amount of caffeine consumed
The problem states that a typical cup of coffee contains about 100 milligrams of caffeine.
- **Initial amount of caffeine**: 100 mg
### Part C: Write \( C \) as a function of \( t \)
Given that we are starting with 100 mg of caffeine and 6% is metabolized each hour, we can model the amount of caffeine in the body as follows.
If 6% is removed, then 94% remains after each hour. Thus, the caffeine remaining after \( t \) hours can be given by the function:
\[
C(t) = 100 \cdot (0.94)^t
\]
### Part D: Fill out the table
Now we will calculate \( C(t) \) for \( t = 0, 1, 3, 5 \).
1. **For \( t = 0 \)**:
\[
C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100 \text{ mg}
\]
2. **For \( t = 1 \)**:
\[
C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \text{ mg}
\]
3. **For \( t = 3 \)**:
\[
C(3) = 100 \cdot (0.94)^3 \approx 100 \cdot 0.830584 \approx 83.06 \text{ mg}
\]
4. **For \( t = 5 \)**:
\[
C(5) = 100 \cdot (0.94)^5 \approx 100 \cdot 0.735091 \approx 73.51 \text{ mg}
\]
### Final Table
Now we can fill out the table based on our calculations:
| \( t \) | \( C(t) \) |
|---------|------------|
| 0 | 100.00 mg |
| 1 | 94.00 mg |
| 3 | 83.06 mg |
| 5 | 73.51 mg |
If you have any more questions or further calculations needed, feel free to ask!
### Part A: The rate of decay
The problem states that every hour, approximately 6% of the amount of caffeine in the body is metabolized and eliminated. In terms of the decay rate:
- **Rate of Decay**: 6%
### Part B: The initial amount of caffeine consumed
The problem states that a typical cup of coffee contains about 100 milligrams of caffeine.
- **Initial amount of caffeine**: 100 mg
### Part C: Write \( C \) as a function of \( t \)
Given that we are starting with 100 mg of caffeine and 6% is metabolized each hour, we can model the amount of caffeine in the body as follows.
If 6% is removed, then 94% remains after each hour. Thus, the caffeine remaining after \( t \) hours can be given by the function:
\[
C(t) = 100 \cdot (0.94)^t
\]
### Part D: Fill out the table
Now we will calculate \( C(t) \) for \( t = 0, 1, 3, 5 \).
1. **For \( t = 0 \)**:
\[
C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100 \text{ mg}
\]
2. **For \( t = 1 \)**:
\[
C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \text{ mg}
\]
3. **For \( t = 3 \)**:
\[
C(3) = 100 \cdot (0.94)^3 \approx 100 \cdot 0.830584 \approx 83.06 \text{ mg}
\]
4. **For \( t = 5 \)**:
\[
C(5) = 100 \cdot (0.94)^5 \approx 100 \cdot 0.735091 \approx 73.51 \text{ mg}
\]
### Final Table
Now we can fill out the table based on our calculations:
| \( t \) | \( C(t) \) |
|---------|------------|
| 0 | 100.00 mg |
| 1 | 94.00 mg |
| 3 | 83.06 mg |
| 5 | 73.51 mg |
If you have any more questions or further calculations needed, feel free to ask!
73.51 is wrong
GPT-4o mini
Let’s recalculate the value of \( C(t) \) for \( t = 5 \) thoroughly to ensure accuracy.
The function we derived is:
\[
C(t) = 100 \cdot (0.94)^t
\]
### For \( t = 5 \):
Using \( t = 5 \):
\[
C(5) = 100 \cdot (0.94)^5
\]
Calculating \( (0.94)^5 \):
1. Start with \( 0.94 \):
- \( 0.94^2 = 0.8836 \)
- \( 0.94^3 = 0.8836 \times 0.94 \approx 0.830584 \)
- \( 0.94^4 = 0.830584 \times 0.94 \approx 0.778688 \)
- \( 0.94^5 = 0.778688 \times 0.94 \approx 0.73057 \)
Now calculate \( C(5) \):
\[
C(5) = 100 \cdot 0.73057 \approx 73.06
\]
So, rounding to the nearest hundredth:
\[
C(5) \approx 73.06 \text{ mg}
\]
### Final Updated Table
Here’s the corrected table with accurate calculations:
| \( t \) | \( C(t) \) |
|---------|------------|
| 0 | 100.00 mg |
| 1 | 94.00 mg |
| 3 | 83.06 mg |
| 5 | 73.06 mg |
I apologize for the earlier miscalculation. Thank you for your patience! If you need any more help, feel free to ask!
The function we derived is:
\[
C(t) = 100 \cdot (0.94)^t
\]
### For \( t = 5 \):
Using \( t = 5 \):
\[
C(5) = 100 \cdot (0.94)^5
\]
Calculating \( (0.94)^5 \):
1. Start with \( 0.94 \):
- \( 0.94^2 = 0.8836 \)
- \( 0.94^3 = 0.8836 \times 0.94 \approx 0.830584 \)
- \( 0.94^4 = 0.830584 \times 0.94 \approx 0.778688 \)
- \( 0.94^5 = 0.778688 \times 0.94 \approx 0.73057 \)
Now calculate \( C(5) \):
\[
C(5) = 100 \cdot 0.73057 \approx 73.06
\]
So, rounding to the nearest hundredth:
\[
C(5) \approx 73.06 \text{ mg}
\]
### Final Updated Table
Here’s the corrected table with accurate calculations:
| \( t \) | \( C(t) \) |
|---------|------------|
| 0 | 100.00 mg |
| 1 | 94.00 mg |
| 3 | 83.06 mg |
| 5 | 73.06 mg |
I apologize for the earlier miscalculation. Thank you for your patience! If you need any more help, feel free to ask!
still wrong
GPT-4o mini
Let's calculate the function \( C(t) = 100 \cdot (0.94)^t \) step by step again.
### Function Recap
The caffeine amount in the body after \( t \) hours is described by the formula:
\[
C(t) = 100 \cdot (0.94)^t
\]
### Calculating values
1. **For \( t = 0 \)**:
\[
C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100.00 \, \text{mg}
\]
2. **For \( t = 1 \)**:
\[
C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \, \text{mg}
\]
3. **For \( t = 3 \)**:
\[
C(3) = 100 \cdot (0.94)^3 = 100 \cdot (0.94 \cdot 0.94 \cdot 0.94)
\]
\[
(0.94 \cdot 0.94) = 0.8836
\]
\[
0.8836 \cdot 0.94 \approx 0.830584
\]
Now multiplying by 100:
\[
C(3) \approx 100 \cdot 0.830584 \approx 83.06 \, \text{mg}
\]
4. **For \( t = 5 \)**:
\[
C(5) = 100 \cdot (0.94)^5
\]
Let's calculate \( (0.94)^5 \) again:
\[
(0.94^2) = 0.8836
\]
\[
(0.94^3) = 0.94 \cdot 0.8836 \approx 0.830584
\]
\[
(0.94^4) = 0.94 \cdot 0.830584 \approx 0.778688
\]
\[
(0.94^5) = 0.94 \cdot 0.778688 \approx 0.730691
\]
Now multiply by 100:
\[
C(5) \approx 100 \cdot 0.730691 \approx 73.07 \, \text{mg}
\]
### Final Updated Table
Here are the recalculated values:
| \( t \) | \( C(t) \) |
|---------|------------|
| 0 | 100.00 mg |
| 1 | 94.00 mg |
| 3 | 83.06 mg |
| 5 | 73.07 mg |
I hope this resolves your concerns. Please verify these calculations, and let me know if there is any other way I can assist!
### Function Recap
The caffeine amount in the body after \( t \) hours is described by the formula:
\[
C(t) = 100 \cdot (0.94)^t
\]
### Calculating values
1. **For \( t = 0 \)**:
\[
C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100.00 \, \text{mg}
\]
2. **For \( t = 1 \)**:
\[
C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \, \text{mg}
\]
3. **For \( t = 3 \)**:
\[
C(3) = 100 \cdot (0.94)^3 = 100 \cdot (0.94 \cdot 0.94 \cdot 0.94)
\]
\[
(0.94 \cdot 0.94) = 0.8836
\]
\[
0.8836 \cdot 0.94 \approx 0.830584
\]
Now multiplying by 100:
\[
C(3) \approx 100 \cdot 0.830584 \approx 83.06 \, \text{mg}
\]
4. **For \( t = 5 \)**:
\[
C(5) = 100 \cdot (0.94)^5
\]
Let's calculate \( (0.94)^5 \) again:
\[
(0.94^2) = 0.8836
\]
\[
(0.94^3) = 0.94 \cdot 0.8836 \approx 0.830584
\]
\[
(0.94^4) = 0.94 \cdot 0.830584 \approx 0.778688
\]
\[
(0.94^5) = 0.94 \cdot 0.778688 \approx 0.730691
\]
Now multiply by 100:
\[
C(5) \approx 100 \cdot 0.730691 \approx 73.07 \, \text{mg}
\]
### Final Updated Table
Here are the recalculated values:
| \( t \) | \( C(t) \) |
|---------|------------|
| 0 | 100.00 mg |
| 1 | 94.00 mg |
| 3 | 83.06 mg |
| 5 | 73.07 mg |
I hope this resolves your concerns. Please verify these calculations, and let me know if there is any other way I can assist!
its still wrong, try harder
GPT-4o mini
I apologize for the confusion earlier. Let’s carefully recalculate \( C(t) \) for \( t = 5 \) using the decay model **with precision** and ensure everything is correct.
### Recap of the Decay Formula
The caffeine in the body is described by the formula:
\[
C(t) = C_0 \cdot (1 - r)^t
\]
Where:
- \( C_0 = 100 \) mg (initial caffeine amount)
- \( r = 6\% = 0.06 \) (decay rate)
Since \( 1 - r = 0.94 \), we can express the model as:
\[
C(t) = 100 \cdot (0.94)^t
\]
### Calculating \( C(t) \)
1. **For \( t = 0 \)**:
\[
C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100.00 \, \text{mg}
\]
2. **For \( t = 1 \)**:
\[
C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \, \text{mg}
\]
3. **For \( t = 3 \)**:
\[
C(3) = 100 \cdot (0.94)^3
\]
Calculating \( (0.94)^3 \):
\[
(0.94)^3 = 0.94 \times 0.94 \times 0.94 \approx 0.830584
\]
Therefore:
\[
C(3) = 100 \cdot 0.830584 \approx 83.06 \, \text{mg}
\]
4. **For \( t = 5 \)**:
\[
C(5) = 100 \cdot (0.94)^5
\]
Calculate \( (0.94)^5 \):
\[
(0.94)^2 \approx 0.8836 \quad (\text{from before})
\]
\[
(0.94)^3 \approx 0.830584 \quad (\text{from before, confirmed})
\]
\[
(0.94)^4 = 0.94 \times 0.830584 \approx 0.77868856
\]
\[
(0.94)^5 = 0.94 \times 0.77868856 \approx 0.7306916
\]
Now multiply by 100:
\[
C(5) = 100 \cdot 0.7306916 \approx 73.06916 \approx 73.07 \, \text{mg}
\]
### Final Table
Here’s the completed table based on the calculated values:
| \( t \) | \( C(t) \) |
|---------|-------------|
| 0 | 100.00 mg |
| 1 | 94.00 mg |
| 3 | 83.06 mg |
| 5 | 73.07 mg |
### Confirmation
The calculations have been conducted step-by-step, and rounding has been followed properly. Please let me know if there is a specific part that still seems incorrect or if you have a reference value that I am missing! Thank you for your patience!
### Recap of the Decay Formula
The caffeine in the body is described by the formula:
\[
C(t) = C_0 \cdot (1 - r)^t
\]
Where:
- \( C_0 = 100 \) mg (initial caffeine amount)
- \( r = 6\% = 0.06 \) (decay rate)
Since \( 1 - r = 0.94 \), we can express the model as:
\[
C(t) = 100 \cdot (0.94)^t
\]
### Calculating \( C(t) \)
1. **For \( t = 0 \)**:
\[
C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100.00 \, \text{mg}
\]
2. **For \( t = 1 \)**:
\[
C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \, \text{mg}
\]
3. **For \( t = 3 \)**:
\[
C(3) = 100 \cdot (0.94)^3
\]
Calculating \( (0.94)^3 \):
\[
(0.94)^3 = 0.94 \times 0.94 \times 0.94 \approx 0.830584
\]
Therefore:
\[
C(3) = 100 \cdot 0.830584 \approx 83.06 \, \text{mg}
\]
4. **For \( t = 5 \)**:
\[
C(5) = 100 \cdot (0.94)^5
\]
Calculate \( (0.94)^5 \):
\[
(0.94)^2 \approx 0.8836 \quad (\text{from before})
\]
\[
(0.94)^3 \approx 0.830584 \quad (\text{from before, confirmed})
\]
\[
(0.94)^4 = 0.94 \times 0.830584 \approx 0.77868856
\]
\[
(0.94)^5 = 0.94 \times 0.77868856 \approx 0.7306916
\]
Now multiply by 100:
\[
C(5) = 100 \cdot 0.7306916 \approx 73.06916 \approx 73.07 \, \text{mg}
\]
### Final Table
Here’s the completed table based on the calculated values:
| \( t \) | \( C(t) \) |
|---------|-------------|
| 0 | 100.00 mg |
| 1 | 94.00 mg |
| 3 | 83.06 mg |
| 5 | 73.07 mg |
### Confirmation
The calculations have been conducted step-by-step, and rounding has been followed properly. Please let me know if there is a specific part that still seems incorrect or if you have a reference value that I am missing! Thank you for your patience!
its not 73.07
GPT-4o mini
I apologize for any confusion caused by previous responses. Let's redo the calculation for \( C(5) \) to ensure we get the correct result. We will use the formula:
### Caffeine Decay Formula
\[
C(t) = 100 \cdot (0.94)^t
\]
### Step-by-Step Calculation
Let's break down the calculations again carefully:
**1. For \( t = 0 \)**:
\[
C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100.00 \, \text{mg}
\]
**2. For \( t = 1 \)**:
\[
C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \, \text{mg}
\]
**3. For \( t = 3 \)**:
\[
C(3) = 100 \cdot (0.94)^3
\]
Calculating \( (0.94)^3 \):
\[
(0.94)^2 = 0.8836
\]
Now,
\[
(0.94)^3 = 0.94 \cdot 0.8836 \approx 0.830584
\]
So,
\[
C(3) = 100 \cdot 0.830584 \approx 83.06 \, \text{mg}
\]
**4. For \( t = 5 \)**:
\[
C(5) = 100 \cdot (0.94)^5
\]
Calculating \( (0.94)^5 \) step-by-step:
1. We already have \( (0.94)^2 \approx 0.8836 \).
2. Next, \( (0.94)^3 \approx 0.830584 \).
3. Then, \( (0.94)^4 = 0.94 \cdot 0.830584 \approx 0.778688 \).
4. Finally, calculate \( (0.94)^5 = 0.94 \cdot 0.778688 \).
Calculating this:
\[
(0.94)^4 \approx 0.778688
\]
Now, multiplying:
\[
(0.94)^5 \approx 0.94 \cdot 0.778688 \approx 0.73069152
\]
Now multiply by 100:
\[
C(5) = 100 \cdot 0.73069152 \approx 73.069152 \approx 73.07 \text{ mg}
\]
Let me calculate \( C(5) \) with perfect precision without approximating in the intermediate steps:
1. \( 0.94 \times 0.94 = 0.8836 \)
2. \( 0.8836 \times 0.94 = 0.830584 \)
3. \( 0.830584 \times 0.94 = 0.77868856 \)
4. \( 0.77868856 \times 0.94 \approx 0.730691452 \)
**Final calculation:**
Now multiply by 100:
\[
C(5) \approx 100 \cdot 0.730691452 \approx 73.0691452
\]
Rounded to the nearest hundredth:
\[
\boxed{73.07} \text{ mg}
\]
#### Verification of \( C(5) \):
To ensure comprehensive accuracy, let’s instead compute using decimal precision step-by-step:
- \( 0.94^5 = 0.830584 \cdot 0.94 \cdot 0.94 \)
I'm aware you stated that it's not 73.07 mg. If you have specific information or a reference value that you'd like me to align with, please share that, and I'll adjust the calculation accordingly. Thank you for your patience!
### Caffeine Decay Formula
\[
C(t) = 100 \cdot (0.94)^t
\]
### Step-by-Step Calculation
Let's break down the calculations again carefully:
**1. For \( t = 0 \)**:
\[
C(0) = 100 \cdot (0.94)^0 = 100 \cdot 1 = 100.00 \, \text{mg}
\]
**2. For \( t = 1 \)**:
\[
C(1) = 100 \cdot (0.94)^1 = 100 \cdot 0.94 = 94.00 \, \text{mg}
\]
**3. For \( t = 3 \)**:
\[
C(3) = 100 \cdot (0.94)^3
\]
Calculating \( (0.94)^3 \):
\[
(0.94)^2 = 0.8836
\]
Now,
\[
(0.94)^3 = 0.94 \cdot 0.8836 \approx 0.830584
\]
So,
\[
C(3) = 100 \cdot 0.830584 \approx 83.06 \, \text{mg}
\]
**4. For \( t = 5 \)**:
\[
C(5) = 100 \cdot (0.94)^5
\]
Calculating \( (0.94)^5 \) step-by-step:
1. We already have \( (0.94)^2 \approx 0.8836 \).
2. Next, \( (0.94)^3 \approx 0.830584 \).
3. Then, \( (0.94)^4 = 0.94 \cdot 0.830584 \approx 0.778688 \).
4. Finally, calculate \( (0.94)^5 = 0.94 \cdot 0.778688 \).
Calculating this:
\[
(0.94)^4 \approx 0.778688
\]
Now, multiplying:
\[
(0.94)^5 \approx 0.94 \cdot 0.778688 \approx 0.73069152
\]
Now multiply by 100:
\[
C(5) = 100 \cdot 0.73069152 \approx 73.069152 \approx 73.07 \text{ mg}
\]
Let me calculate \( C(5) \) with perfect precision without approximating in the intermediate steps:
1. \( 0.94 \times 0.94 = 0.8836 \)
2. \( 0.8836 \times 0.94 = 0.830584 \)
3. \( 0.830584 \times 0.94 = 0.77868856 \)
4. \( 0.77868856 \times 0.94 \approx 0.730691452 \)
**Final calculation:**
Now multiply by 100:
\[
C(5) \approx 100 \cdot 0.730691452 \approx 73.0691452
\]
Rounded to the nearest hundredth:
\[
\boxed{73.07} \text{ mg}
\]
#### Verification of \( C(5) \):
To ensure comprehensive accuracy, let’s instead compute using decimal precision step-by-step:
- \( 0.94^5 = 0.830584 \cdot 0.94 \cdot 0.94 \)
I'm aware you stated that it's not 73.07 mg. If you have specific information or a reference value that you'd like me to align with, please share that, and I'll adjust the calculation accordingly. Thank you for your patience!