Question

Rate of decay is 6 %
Amount of coffee initially is 100mg
Let C represent the amount of coffee in the body and t represent the number of hours since a cup of coffee was consumed
C(t)=
Answer
Write C as a function of t
Answer
Fill in table
t 5
C(t)
Answer
Both are wrong
Answer
I know they wrong cause they marked wrong but here’s the problem again
A cup of coffee contains about 100 mg of caffeine. Every hour approximately 6% of the amount of caffeine in the body is metabolized and eliminated
What is the C(t)=
And the table
t 0 1 3 5
C(t)
Answered by GPT-4o mini
To model the amount of coffee remaining in the body over time, we can use the exponential decay formula:

\[ C(t) = C_0 \cdot e^{-kt} \]

Where:
- \( C_0 \) is the initial amount of coffee,
- \( k \) is the decay constant,
- \( t \) is the time in hours,
- \( e \) is the base of the natural logarithm.

In your case:
- The initial amount of coffee \( C_0 = 100 \) mg,
- The decay rate is 6%, which means that after 1 hour, you will have 94% of the coffee remaining (100% - 6% = 94%).

This leads us to calculate the decay constant \( k \):

1. Since 94% of the coffee remains after 1 hour:
\[
C(1) = 0.94 \cdot C_0 = 0.94 \cdot 100 = 94 \text{ mg}
\]

2. Therefore, we can express this with the decay formula for \( t = 1 \):
\[
C(1) = C_0 \cdot e^{-k \cdot 1}
\]
Substituting the numbers:
\[
94 = 100 \cdot e^{-k}
\]
Dividing both sides by 100:
\[
0.94 = e^{-k}
\]
Taking the natural logarithm:
\[
-k = \ln(0.94)
\]
So:
\[
k = -\ln(0.94)
\]

3. Calculating \( k \):
\[
k \approx -(-0.0619) \approx 0.0619
\]

4. Finally, substituting \( C_0 \) and \( k \) back into the exponential decay formula gives us:
\[
C(t) = 100 \cdot e^{-0.0619t}
\]

Thus, the amount of coffee in the body at time \( t \) hours since consumption is:

\[
C(t) = 100 \cdot e^{-0.0619t}
\]
Answered by GPT-4o mini
Based on the calculation we performed earlier, the amount of coffee \( C(t) \) as a function of time \( t \) (in hours) is given by:

\[
C(t) = 100 \cdot e^{-0.0619t}
\]

Here, \( C(t) \) represents the amount of coffee (in mg) remaining in the body \( t \) hours after consumption, where:
- \( 100 \) mg is the initial amount of coffee,
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828),
- \( -0.0619 \) is the decay constant reflecting a decay rate of 6% per hour,
- \( t \) is the number of hours since the coffee was consumed.

You can use this function to calculate the remaining amount of coffee in the body for any given time \( t \).
Answered by GPT-4o mini
To find the amount of coffee remaining in the body after \( t = 5 \) hours using the formula

\[
C(t) = 100 \cdot e^{-0.0619t}
\]

we will substitute \( t = 5 \) into the equation:

1. Calculate \( e^{-0.0619 \cdot 5} \):

\[
-0.0619 \cdot 5 = -0.3095
\]

Now, calculate \( e^{-0.3095} \):

\[
e^{-0.3095} \approx 0.7345 \quad (\text{approximate value})
\]

2. Substitute this value back into the equation for \( C(t) \):

\[
C(5) = 100 \cdot 0.7345 \approx 73.45 \text{ mg}
\]

So for \( t = 5 \):

\[
C(5) \approx 73.45 \text{ mg}
\]

You can now fill in your table as follows:

| \( t \) | \( C(t) \) |
|:----------:|:-------------:|
| 5 | 73.45 mg |
Answered by GPT-4o mini
I apologize for the confusion in my previous calculations. Let's carefully recalculate the amount of coffee remaining after 5 hours, using the decay model \( C(t) = 100 \cdot e^{-0.0619t} \).

### Step-by-step Calculation:

1. **Given Data:**
- Initial amount \( C_0 = 100 \) mg
- Decay constant \( k \approx 0.0619 \)
- Time \( t = 5 \) hours

2. **Apply the formula:**
\[
C(t) = 100 \cdot e^{-0.0619t}
\]
Substituting \( t = 5 \):
\[
C(5) = 100 \cdot e^{-0.0619 \cdot 5}
\]

3. **Calculate the exponent:**
\[
-0.0619 \cdot 5 = -0.3095
\]

4. **Calculate \( e^{-0.3095} \):**
Using a calculator or software:
\[
e^{-0.3095} \approx 0.7345
\]

5. **Calculate \( C(5) \):**
\[
C(5) = 100 \cdot 0.7345 \approx 73.45 \text{ mg}
\]

This value is indeed \( C(5) \approx 73.45 \) mg.

### Summary Table:
Here’s the completed table entry:

| \( t \) | \( C(t) \) |
|---------|--------------|
| 5 | 73.45 mg |

If you believe these calculations are still incorrect or if there are any specific aspects that should be addressed differently, please provide more details, and I'll be glad to assist further!
Answered by GPT-4o mini
Thank you for your patience and for clarifying the problem. Since 6% of the caffeine is metabolized every hour, this means that after 1 hour, 94% of the caffeine remains in the body. We can express the decay equation in a more straightforward way, specifically related to the percentage remaining.

### Formulating \( C(t) \)

If \( C_0 = 100 \) mg (the initial amount of caffeine), then after each hour, the amount of caffeine remaining can be determined using the formula:

\[
C(t) = C_0 \times (1 - r)^t
\]

where:
- \( C_0 = 100 \) mg (initial caffeine),
- \( r = 0.06 \) (decay rate of 6%),
- \( (1 - r) = 0.94 \) (percentage of caffeine remaining).

Thus, the function becomes:

\[
C(t) = 100 \times 0.94^t
\]

### Calculating Values for the Table

Now we can calculate \( C(t) \) for the specified values of \( t \):

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 \( t = 3 \)**:
\[
C(3) = 100 \times 0.94^3 \approx 100 \times 0.8306 \approx 83.06 \text{ mg}
\]

4. **For \( t = 5 \)**:
\[
C(5) = 100 \times 0.94^5 \approx 100 \times 0.7351 \approx 73.51 \text{ mg}
\]

### Summary Table

Now let's compile the results into the table:

| \( t \) | \( C(t) \) |
|---------|-------------|
| 0 | 100 mg |
| 1 | 94 mg |
| 3 | 83.06 mg |
| 5 | 73.51 mg |

This set of calculations should now be accurately reflecting the decay of caffeine in the body at the specified time intervals. Thank you for your understanding, and please let me know if you have any further questions!