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Chapter 9: Problem 67

A dextrose solution is being administered to a patient intravenously. The 1-liter (L) bottle holding the solution is removed and replaced by another as soon as the contents drop to approximately \(5 \%\) of the initial (1-L) amount. The rate of discharge is constant, and it takes \(6 \mathrm{hr}\) to discharge \(95 \%\) of the contents of a full bottle. Draw a graph showing the amount of dextrose solution in a bottle in the IV system over a 24 -hr period, assuming that we started with a full bottle.

Short Answer

Expert verified
The amount of dextrose solution in the IV system over a 24-hour period can be plotted by first calculating the discharge rate per hour, which is \(0.1583 L/hour\). Next, create a table to record the amount of dextrose solution for each hour, taking into account the replacement of the bottle when only \(5\%\) of the initial amount is left. Finally, plot this data on a graph that shows a step-like pattern with decreasing curves at each step, representing the discharge of the dextrose solution over the 24-hour period.

Step by step solution

01

The rate of discharge is constant, and it takes 6 hours to discharge \(95\%\) of the 1-liter bottle. We need to find out how much solution is discharged per hour. To do this, we calculate: \[ Discharge\;rate = \frac{95\% \cdot 1\;L}{6\;hours}\] #Step 2: Calculate the discharge rate per hour #

Now, let's calculate the discharge rate per hour: \[ Discharge\;rate = \frac{0.95\;L}{6\;hours} = 0.1583\;L/hour\] #Step 3: Determine the solution level for each hour over a 24-hour period#
02

We will now create a table to record the amount of dextrose solution remaining in the bottle for each hour over a 24-hour period. We will take into account that when the bottle has only \(5\%\) of the initial amount left, it is replaced by a full bottle. | Hour | Amount of dextrose solution in the bottle (in L)| |------|-----------------------------------------------| | 0 | 1 | | 1 | 1 - 0.1583 | | 2 | 1 - 2(0.1583) | | 3 | 1 - 3(0.1583) | | 4 | 1 - 4(0.1583) | | 5 | 1 - 5(0.1583) | | 6 | 1 (new bottle) | | 7 | 1 - 0.1583 | | ... | ... | | 24 | ... | #Step 4: Plot the graph#

Now, we can plot the amount of dextrose solution in the bottle (y-axis) against time, in hours (x-axis), based on the data in the table. To do this, we will draw a decreasing curve from 1 liter to \(5\%\) remaining at each 6-hour interval, at which point the curve will jump back to 1 liter and start decreasing again. The final graph should show a step-like pattern, with a decreasing curve for each step, representing the discharge of the dextrose solution over a 24-hour period.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate of Discharge
In this context, the rate of discharge refers to how quickly the dextrose solution is being used or removed from the IV bottle. The problem states that it takes 6 hours to empty 95% of a full 1-liter bottle. This means the discharge rate is steady and continuous over this period.

To calculate the rate of discharge, we first convert the 95% into liters, which is 0.95 liters. We divide this by the time it takes, which is 6 hours. This calculation gives us the discharge rate per hour:\[ Discharge\;rate = \frac{0.95\;L}{6\;hours} = 0.1583\;L/hour \]
This constant rate means that every hour, approximately 0.1583 liters of dextrose solution is administered to the patient.
Graphical Representation
Creating a visual representation of how the dextrose solution is used over time helps in understanding the pattern of discharge. In this case, we are asked to plot the amount of solution over a 24-hour period.

The graph's x-axis will represent time in hours, while the y-axis will represent the amount of solution remaining in the bottle in liters. The plot will show a series of steps:
  • Starting at 1 liter, the amount will decrease linearly over 6 hours to approximately 0.05 liters.
  • Once the solution reaches this level, the bottle is refilled to 1 liter.
This step-wise pattern will repeat every 6 hours as new bottles are introduced. This graphical approach helps in visualizing the pattern of discharge clearly.
Step Function
A step function is a piecewise constant function that changes values at certain points, creating a step-like pattern in a graph. In this exercise, the amount of solution in the bottle over time can be modeled as a step function.

Every 6 hours, the bottle is essentially reset:
  • It decreases steadily from 1 liter to 0.05 liters over these 6 hours.
  • At the 6th hour mark, the amount jumps back to 1 liter, as a new bottle is used.
This pattern creates a series of descending steps when graphed over a 24-hour period, providing a clear visual of the reoccurring drops and resets in solution level.
Dextrose Solution
Dextrose solutions are commonly used in medical settings as intravenous fluids. They provide essential nutrients in the form of glucose, which the body uses for energy. Such solutions are often prescribed when a patient cannot eat or drink adequately due to illness or surgery.

The concentration of the solution can vary, but in this exercise, we are looking at a standard 1-liter bottle that is being used consistently over time. Understanding the rate at which these solutions are discharged helps in administering the correct dosage and ensures that the patient receives a continuous supply of nutrients.

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