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Sagot :
Let's approach this problem step-by-step to determine how many milliliters of normal saline need to be added to achieve the desired final concentration.
1. Determine the initial concentration of verapamil in mg/mL:
- We are given that the concentration is \( \frac{5 \text{ mg}}{2 \text{ mL}} \).
- Converting this to mg/mL, we get:
[tex]\[ \text{Initial concentration} = \frac{5 \text{ mg}}{2 \text{ mL}} = 2.5 \text{ mg/mL} \][/tex]
2. Calculate the total amount of verapamil in the vial:
- The volume of the verapamil solution is \( 4 \text{ mL} \).
- Using the concentration, the total amount is:
[tex]\[ \text{Total amount of verapamil} = 4 \text{ mL} \times 2.5 \text{ mg/mL} = 10 \text{ mg} \][/tex]
3. Determine the desired final concentration of the solution:
- It is given as \( 0.1 \text{ mg/mL} \).
4. Calculate the total volume needed to achieve the final concentration:
- We need to find out how much solution will give us the final concentration, knowing the total amount of verapamil does not change.
- Let \( V_{\text{final}} \) represent the total volume needed.
- The equation to find \( V_{\text{final}} \) is:
[tex]\[ \text{Total amount of verapamil} = \text{Desired concentration} \times V_{\text{final}} \][/tex]
Substituting the known values:
[tex]\[ 10 \text{ mg} = 0.1 \text{ mg/mL} \times V_{\text{final}} \][/tex]
Solving for \( V_{\text{final}} \):
[tex]\[ V_{\text{final}} = \frac{10 \text{ mg}}{0.1 \text{ mg/mL}} = 100 \text{ mL} \][/tex]
5. Calculate how much saline needs to be added:
- The final volume required is \( 100 \text{ mL} \).
- Initially, we have \( 4 \text{ mL} \) of verapamil solution.
- The volume of saline to be added is:
[tex]\[ \text{Volume of saline} = V_{\text{final}} - \text{Initial volume of verapamil} \][/tex]
[tex]\[ \text{Volume of saline} = 100 \text{ mL} - 4 \text{ mL} = 96 \text{ mL} \][/tex]
So, [tex]\( \boxed{96} \)[/tex] milliliters of normal saline need to be added to the [tex]\( 4 \text{ mL} \)[/tex] vial of verapamil to achieve the desired final concentration of [tex]\( 0.1 \text{ mg/mL} \)[/tex].
1. Determine the initial concentration of verapamil in mg/mL:
- We are given that the concentration is \( \frac{5 \text{ mg}}{2 \text{ mL}} \).
- Converting this to mg/mL, we get:
[tex]\[ \text{Initial concentration} = \frac{5 \text{ mg}}{2 \text{ mL}} = 2.5 \text{ mg/mL} \][/tex]
2. Calculate the total amount of verapamil in the vial:
- The volume of the verapamil solution is \( 4 \text{ mL} \).
- Using the concentration, the total amount is:
[tex]\[ \text{Total amount of verapamil} = 4 \text{ mL} \times 2.5 \text{ mg/mL} = 10 \text{ mg} \][/tex]
3. Determine the desired final concentration of the solution:
- It is given as \( 0.1 \text{ mg/mL} \).
4. Calculate the total volume needed to achieve the final concentration:
- We need to find out how much solution will give us the final concentration, knowing the total amount of verapamil does not change.
- Let \( V_{\text{final}} \) represent the total volume needed.
- The equation to find \( V_{\text{final}} \) is:
[tex]\[ \text{Total amount of verapamil} = \text{Desired concentration} \times V_{\text{final}} \][/tex]
Substituting the known values:
[tex]\[ 10 \text{ mg} = 0.1 \text{ mg/mL} \times V_{\text{final}} \][/tex]
Solving for \( V_{\text{final}} \):
[tex]\[ V_{\text{final}} = \frac{10 \text{ mg}}{0.1 \text{ mg/mL}} = 100 \text{ mL} \][/tex]
5. Calculate how much saline needs to be added:
- The final volume required is \( 100 \text{ mL} \).
- Initially, we have \( 4 \text{ mL} \) of verapamil solution.
- The volume of saline to be added is:
[tex]\[ \text{Volume of saline} = V_{\text{final}} - \text{Initial volume of verapamil} \][/tex]
[tex]\[ \text{Volume of saline} = 100 \text{ mL} - 4 \text{ mL} = 96 \text{ mL} \][/tex]
So, [tex]\( \boxed{96} \)[/tex] milliliters of normal saline need to be added to the [tex]\( 4 \text{ mL} \)[/tex] vial of verapamil to achieve the desired final concentration of [tex]\( 0.1 \text{ mg/mL} \)[/tex].
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