Sam takes 800 mg ibuprofen for her nagging backache every hour that the ibuprofen is in her system a quarter 25% of the medicine dissolves in her body write a function that models the amount of ibuprofen in Sam’s body over time he’s asked for hours and wife for the amount of ibuprofen in milligrams remaining in the body

Answer 1

To model the amount of ibuprofen in Sam's body over time, we can use an exponential decay function. Given that Sam takes 800 mg of ibuprofen every hour and that 25% (or \(0.25\)) of the medicine dissolves in her body each hour, we have the following details to consider:

1. After the first hour, 25% of the ibuprofen remains.
2. This means 75% of the ibuprofen is lost or has dissolved.

This translates into the following amount of ibuprofen remaining after each hour:

• After 1 hour, the amount remaining is \(800 \times (1 - 0.25) = 800 \times 0.75 = 600 \text{ mg}\).
• After 2 hours, the amount remaining is \(600 \times 0.75 = 450 \text{ mg}\).
• After 3 hours, the amount remaining is \(450 \times 0.75 = 337.5 \text{ mg}\), and so on.

We can derive a function that models this scenario. Let:

• \( t \) be the number of hours after taking ibuprofen, and
• \( A(t) \) be the amount of ibuprofen in milligrams remaining in her body.

The function can be expressed as follows:

\[ A(t) = 800 \times (0.75)^t \]

However, since Sam takes 800 mg every hour, we need to account for the ibuprofen taken at each hour. Therefore, the total amount of ibuprofen in her system at any hour can be modeled using a piecewise function:

1. For \( t = 0 \): \( A(0) = 800 \) mg (the initial dose).
2. For \( t = 1 \): \( A(1) = 800 + 800 \times 0.75^1 = 800 + 600 = 1400 \) mg.
3. For \( t = 2 \): \( A(2) = 800 + 800 \times 0.75^1 + 800 \times 0.75^2 = 800 + 600 + 450 = 1850 \) mg.
4. For \( t = 3 \): \( A(3) = 800 + 800 \times 0.75^1 + 800 \times 0.75^2 + 800 \times 0.75^3 \), etc.

In general, the equation for the total amount of ibuprofen in Sam's body after \( t \) hours can be expressed with a summation:

\[ A(t) = 800 \sum_{i=0}^{t} (0.75)^i \]

This summation can be calculated using the formula for the sum of a geometric series:

\[ \sum_{i=0}^{n} r^i = \frac{1 - r^{n+1}}{1 - r} \]

Where in our case, \( r = 0.75 \) and \( n = t \).

Thus,

\[ A(t) = 800 \times \frac{1 - (0.75)^{t+1}}{1 - 0.75} = 800 \times \frac{1 - (0.75)^{t+1}}{0.25} = 3200 \times (1 - (0.75)^{t+1}) \]

This function can be used to determine the total amount of ibuprofen in Sam's body at any time \( t \) hours after her initial dose.

Answer 2

800( )^x

Answer 3

To express the amount of ibuprofen in Sam's body using the form \(800( \text{something} )^x\), we can model the decay of the ibuprofen taken each hour while also considering how much is added each hour.

However, to match your request directly to the exponential form with a focus on decay, we will consider only the decay of the ibuprofen that remains in her system without adding new doses within each hour.

Considering only the decay:

1. Each hour, 75% of the previous amount remains. If we denote \(x\) as the number of hours, then the amount of ibuprofen remaining from just the initial dose can be expressed as:

\[ A(x) = 800 \cdot (0.75)^x \]

Where:

• \(800\) is the initial amount of ibuprofen taken.
• \(0.75\) is the fraction of ibuprofen that remains after dissolving.
• \(x\) is the number of hours since the ibuprofen was taken.

However, keep in mind that this equation does not account for the additional doses that are taken each hour.

If you want to consider the total amount of ibuprofen in her body after each dose over time:

To include the new dosage of 800 mg every hour, we would typically sum the amounts from previous doses.

Total Amount (Including New Doses):

The total amount of ibuprofen in Sam’s body at any hour \(x\) (considering new doses) can become complex but here is a simplified view using \(800( \text{something} )^{x}\):

\[ A(x) = 800 \cdot \left( (0.75)^x + (0.75)^{x-1} + (0.75)^{x-2} + \ldots + (0.75)^0 \right) \]

While we can keep the factor of 800, the actual form you requested wouldn’t simplify into a clean \(800( \text{something} )^x\) because of the cumulative nature of the doses.

However, if we limit to just the decay of the first 800 mg, then the function can simply be expressed as:

\[ A(x) = 800 \cdot (0.75)^x \]

In a summary:

• For just the initial dose decaying: \(800(0.75)^x\)
• For cumulative doses over time: \(A(x) = 800 \cdot \frac{1 - (0.75)^{x+1}}{0.25} = 3200(1 - (0.75)^{x+1})\)

This describes the total amount of ibuprofen remaining considering it keeps getting replenished every hour.

Answer 4

When there is less than 200 mg of medicine left in the body, it is safe to take again. How long will it take before Sam can take more medicine

Answer 5

To determine when Sam can safely take more ibuprofen, we need to find the point in time when the amount of ibuprofen in her body falls below 200 mg.

Using the decay function based on the initial dose and the decay factor:

\[ A(x) = 800 \cdot (0.75)^x \]

We set the equation equal to 200 mg to find the time \(x\):

\[ 800 \cdot (0.75)^x < 200 \]

To solve for \(x\), we first divide both sides by 800:

\[ (0.75)^x < \frac{200}{800} \]

Simplifying the right side, we get:

\[ (0.75)^x < 0.25 \]

Next, we take the logarithm of both sides. You can use either the natural logarithm (ln) or base 10 logarithm (log):

\[ \log((0.75)^x) < \log(0.25) \]

Using the properties of logarithms, we can bring down the exponent:

\[ x \cdot \log(0.75) < \log(0.25) \]

Now, we can solve for \(x\). Note that \(\log(0.75)\) is a negative number, so when we divide by it, we need to reverse the inequality:

\[ x > \frac{\log(0.25)}{\log(0.75)} \]

Now we will calculate this value using a calculator:

1. \(\log(0.25) \approx -0.60206\) (base 10)
2. \(\log(0.75) \approx -0.12494\) (base 10)

Now, substituting these values:

\[ x > \frac{-0.60206}{-0.12494} \approx 4.81 \]

Since \(x\) must be a whole number (as Sam takes medicine every hour), we round it up:

\[ x \geq 5 \]

Therefore, it will take approximately 5 hours before Sam can safely take more ibuprofen.