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An IV administers medication to a patient's bloodstream at a rate of 3 cubic centimeters per hour. At the same time, the patient's organs remove the medication from the patient's bloodstream at a rate proportional to the current volume V of medication in the bloodstream. Which equation describes this relationship? Choose 1 answer: (A) (dV)/(dt)=3k-V (B) (dV)/(dt)=3-kV (c) (dV)/(dt)=-3kV (D) (dV)/(dt)=k-3V

An IV administers medication to a patient's bloodstream at a rate of 33 cubic centimeters per hour.\newlineAt the same time, the patient's organs remove the medication from the patient's bloodstream at a rate proportional to the current volume V V of medication in the bloodstream.\newlineWhich equation describes this relationship?\newlineChoose 11 answer:\newline(A) dVdt=3k−V \frac{d V}{d t}=3 k-V \newline(B) dVdt=3−kV \frac{d V}{d t}=3-k V \newline(c) dVdt=−3kV \frac{d V}{d t}=-3 k V \newline(D) dVdt=k−3V \frac{d V}{d t}=k-3 V

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Q. An IV administers medication to a patient's bloodstream at a rate of 33 cubic centimeters per hour.\newlineAt the same time, the patient's organs remove the medication from the patient's bloodstream at a rate proportional to the current volume V V of medication in the bloodstream.\newlineWhich equation describes this relationship?\newlineChoose 11 answer:\newline(A) dVdt=3k−V \frac{d V}{d t}=3 k-V \newline(B) dVdt=3−kV \frac{d V}{d t}=3-k V \newline(c) dVdt=−3kV \frac{d V}{d t}=-3 k V \newline(D) dVdt=k−3V \frac{d V}{d t}=k-3 V
  1. Understand the problem: Understand the problem.\newlineThe IV administers medication at a constant rate of 33 cubic centimeters per hour. At the same time, the patient's organs are removing the medication at a rate proportional to the current volume of medication in the bloodstream. We need to find an equation that represents this situation.
  2. Set up the differential equation: Set up the differential equation.\newlineThe rate of change of the volume of medication in the bloodstream, dVdt\frac{dV}{dt}, is equal to the rate of medication being administered minus the rate at which the medication is being removed by the patient's organs.
  3. Translate into mathematical terms: Translate the information into mathematical terms.\newlineThe rate of administration is a constant 33 cubic centimeters per hour, so this is a positive term in our equation. The rate of removal is proportional to the volume VV, which means it will be a negative term since it is being removed. The constant of proportionality is kk. Therefore, the equation should have a positive term for administration and a negative term for removal.
  4. Identify the correct equation: Identify the correct equation.\newlineBased on the information given, the equation should look like this: (dVdt)=rate of administration−rate of removal(\frac{dV}{dt}) = \text{rate of administration} - \text{rate of removal}. The rate of administration is 33, and the rate of removal is proportional to VV, so it should be kVkV. Therefore, the equation should be (dVdt)=3−kV(\frac{dV}{dt}) = 3 - kV.
  5. Match with given choices: Match the equation with the given choices.\newlineThe correct equation from the choices provided that matches our equation from Step 44 is (B) (dVdt=3−kV)(\frac{dV}{dt} = 3 - kV).

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