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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)

11 answers

Let's go through each part of the problem step by step.

Part A: Determine the rate of decay

Given that 6% of the caffeine in the body is metabolized and eliminated every hour, the remaining amount of caffeine after one hour is 100% - 6% = 94%.

Thus, the rate of decay is 6% per hour.

Part B: Initial amount of caffeine

According to the problem, the initial amount of caffeine consumed in this scenario is 100 mg.

Part C: Write \(C(t)\) as a function of \(t\)

We know that the initial amount \(C(0)\) is 100 mg, and after each hour, 94% of the caffeine remains. This results in an exponential decay function: \[ C(t) = C(0) \times (1 - \text{decay rate})^t \] Plugging in the values we have: \[ C(t) = 100 \times (0.94)^t \]

Part D: Fill out the table

Now, we can calculate \(C(t)\) for \(t = 0\), \(t = 1\), \(t = 3\), and \(t = 5\).

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

  2. For \(t = 1\): \[ C(1) = 100 \times (0.94)^1 = 100 \times 0.94 = 94.00 \text{ mg} \]

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

  4. For \(t = 5\): \[ C(5) = 100 \times (0.94)^5 \approx 100 \times 0.735091 = 73.51 \text{ mg} \]

Final table

Here are the values for \(C(t)\) rounded to the nearest hundredth:

| t | C(t) | |:-------:|:----------:| | 0 | 100.00 | | 1 | 94.00 | | 3 | 83.06 | | 5 | 73.51 |

In summary, the answers are:

  • Part A: 6%
  • Part B: 100 mg
  • Part C: \(C(t) = 100 \times (0.94)^t\)
  • Part D:
    • \(C(0) = 100.00\)
    • \(C(1) = 94.00\)
    • \(C(3) = 83.06\)
    • \(C(5) = 73.51\)
Part C is Wrong

Let’s re-evaluate Part C carefully.

We are given:

  • The initial amount of caffeine consumed: 100 mg
  • The decay is such that 6% of caffeine is metabolized every hour, meaning that 94% remains after each hour.

We express the remaining caffeine functionally based on time \(t\) (in hours).

Starting with the facts:

  • The initial amount of caffeine is \( C(0) = 100 \) mg.
  • After each hour, \(C(t)\) shrinks to \(94%\) of the previous amount.

This means: \[ C(t) = C(0) \times (0.94)^t \]

Substituting \(C(0) = 100\) mg gives us: \[ C(t) = 100 \times (0.94)^t \]

Part C is correctly represented by this formula.

Could you clarify what you believe is incorrect, or if you would like it in a different format? If you need more assistance, feel free to ask!

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) =

Let’s clearly define the function \(C(t)\) for the amount of caffeine in the body in milligrams after \(t\) hours since consuming a cup of coffee.

Given the information:

  • The initial caffeine amount is \(C(0) = 100\) mg.
  • Each hour, 6% of the caffeine is metabolized, which means that 94% of it remains.

The formula for caffeine remaining after \(t\) hours can be expressed as:

\[ C(t) = C(0) \cdot (0.94)^t \]

Substituting in the initial amount:

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

Thus, the final answer for Part C is:

C(t) = 100 \cdot (0.94)^t

This function indicates the amount of caffeine remaining in the body after \(t\) hours.

Write that in function form.

Certainly! In function notation, you would write the caffeine function as follows:

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

Where:

  • \( C(t) \) represents the amount of caffeine in the body in milligrams,
  • \( t \) represents the number of hours since a cup of coffee was consumed.

This is the precise function form you can use for your requirements.

U Sure???????????????

Yes, I confirm that the function for the amount of caffeine remaining in the body, given the information provided, is correctly expressed as:

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

To clarify:

  • 100 is the initial amount of caffeine in milligrams (mg).
  • 0.94 is the proportion of caffeine remaining after one hour (since 6% is metabolized).
  • t is the time in hours after consumption.

This equation represents an exponential decay model, which is suitable for describing how a substance decreases in the body over time due to metabolism.

If you have any specific concerns or if there's an aspect you'd like to revise, please let me know!

The bot is correct its 100(0.94)^t

Thank you for your message! I'm glad we could clarify that the function is indeed:

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

This accurately describes the amount of caffeine remaining in the body after \(t\) hours. If you have any more questions or need further assistance with any other topics, feel free to ask!