Problem 9
Question
A \(50 \mathrm{mg}\) dose of quinine is given to a patient to prevent malaria. Quinine leaves the body at a rate of \(6 \%\) per hour. (a) Find a formula for the amount, \(A\) (in \(\mathrm{mg}\) ), of quinine in the body \(t\) hours after the dose is given. (b) How much quinine is in the body after 24 hours? (c) Graph \(A\) as a function of \(t\). (d) Use the graph to estimate when \(5 \mathrm{mg}\) of quinine remains.
Step-by-Step Solution
Verified Answer
(a) \(A(t) = 50 \times 0.94^t\); (b) \(9.96\,\text{mg}\); (c) Graph \(A(t)\); (d) Approximately 54 hours.
1Step 1: Set up the decay formula
We start by deriving the formula for the amount of quinine in the body over time. Since quinine decays at 6% per hour, the amount remaining after each hour is 94% of the amount in the previous hour. This can be expressed as an exponential decay formula. The general formula for exponential decay is:\[ A(t) = A_0 \times (1 - r)^t \]where \( A_0 \) is the initial amount, \( r \) is the decay rate as a decimal, and \( t \) is the time in hours.
2Step 2: Plug values into the decay formula
Given that the initial amount \(A_0\) is 50 mg and the decay rate \(r\) is 0.06, we can substitute these values into the formula to find:\[ A(t) = 50 \times (1 - 0.06)^t = 50 \times 0.94^t \] This formula will give us the amount of quinine in the body \(t\) hours after administration.
3Step 3: Calculate the amount after 24 hours
To find how much quinine remains after 24 hours, we substitute \(t = 24\) into the derived formula:\[ A(24) = 50 \times 0.94^{24} \] Calculate this using a calculator to find the amount after 24 hours.
4Step 4: Graph the function
Graph the function \( A(t) = 50 \times 0.94^t \) using suitable software or graphing tool. Plot the points for a range of \( t \) values to illustrate how the amount of quinine decreases over time. The x-axis will represent time in hours and the y-axis will represent the quinine amount in mg.
5Step 5: Estimate when 5 mg remains
Using the graph plotted in the previous step, determine the time at which the quinine level reaches 5 mg. Look at the graph and find when the curve intersects the horizontal line at \( A(t) = 5 \). This is an estimation of the time it takes for the quinine to reduce to 5 mg.
Key Concepts
Quinine EliminationExponential FunctionGraphing Functions
Quinine Elimination
When you take a medication like quinine, understanding how it leaves the body over time can be crucial. Quinine is a medication used primarily for treating malaria and can have potentially strong effects. Quinine elimination from the body happens at a specific rate. In this scenario, quinine leaves the body at a rate of 6% per hour.
This means that every hour, only 94% of the quinine remains in your system. This can be calculated using an exponential decay formula. It's important to remember that this type of decay is consistent; the same percentage is removed each hour.
For quinine elimination, this ensures that the medication decreases in a predictable manner, making it easier for healthcare professionals to predict when a patient might need another dose or when quinine levels will reach a certain threshold.
This means that every hour, only 94% of the quinine remains in your system. This can be calculated using an exponential decay formula. It's important to remember that this type of decay is consistent; the same percentage is removed each hour.
For quinine elimination, this ensures that the medication decreases in a predictable manner, making it easier for healthcare professionals to predict when a patient might need another dose or when quinine levels will reach a certain threshold.
Exponential Function
An exponential function is a type of mathematical formula that describes processes where quantities decrease or increase at rates proportional to their current value. The general format is:
\[ A(t) = A_0 \times (1 - r)^t \]
where:
In the context of quinine, the initial amount (\( A_0 \)) is 50 mg, and the decay rate is 0.06 per hour. The exponential decay formula for the quinine is \[ A(t) = 50 \times 0.94^t \] indicating how the medication decreases with time.
This type of function is instrumental because it helps make predictions about the remaining medication in the body, aiding in treatment planning.
\[ A(t) = A_0 \times (1 - r)^t \]
where:
- \( A_0 \) is the initial amount.
- \( r \) is the rate of change (here, it's the decay rate, expressed as a decimal).
- \( t \) represents time.
In the context of quinine, the initial amount (\( A_0 \)) is 50 mg, and the decay rate is 0.06 per hour. The exponential decay formula for the quinine is \[ A(t) = 50 \times 0.94^t \] indicating how the medication decreases with time.
This type of function is instrumental because it helps make predictions about the remaining medication in the body, aiding in treatment planning.
Graphing Functions
Graphing an exponential function like the one describing quinine elimination is a visual method to understand how the medication changes over time.
For the function \( A(t) = 50 \times 0.94^t \), you can notice the general shape of an exponential decay curve. This curve decreases faster at the beginning and then tapers off slowly, approaching but never reaching zero.
To create this graph, you plot time \( t \) on the x-axis and the amount of quinine \( A(t) \) on the y-axis. This visualization helps determine key points like when there's 5 mg left. By following the curve until it intersects with the horizontal line representing 5 mg, you can make informed decisions on medication schedules or necessary adjustments in dosage. Graphing functions not only supports theoretical findings but also allows for practical applications, making it a vital skill in understanding and managing medication effects.
With computer software or graphing tools, achieving an accurate representation of this curve becomes effortless, providing insights at a glance.
For the function \( A(t) = 50 \times 0.94^t \), you can notice the general shape of an exponential decay curve. This curve decreases faster at the beginning and then tapers off slowly, approaching but never reaching zero.
To create this graph, you plot time \( t \) on the x-axis and the amount of quinine \( A(t) \) on the y-axis. This visualization helps determine key points like when there's 5 mg left. By following the curve until it intersects with the horizontal line representing 5 mg, you can make informed decisions on medication schedules or necessary adjustments in dosage. Graphing functions not only supports theoretical findings but also allows for practical applications, making it a vital skill in understanding and managing medication effects.
With computer software or graphing tools, achieving an accurate representation of this curve becomes effortless, providing insights at a glance.
Other exercises in this chapter
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