A chemical is diluted out of a tank by pumping pure water into the tank and pumping the existing solution out of it, so the volume at any time t is 20+2t. The amount z of chemical in the tank decreases at a rate proportional to z and inversely proportional to the volume of solution in the tank. Which equation describes this relationship? Choose 1 answer: (A) (dz)/(dt)=-(k(20+2t))/(z) (B) (dz)/(dt)=kz-(1)/(20+2t) (C) (dz)/(dt)=k(20+2t)-(1)/(z) (D) (dz)/(dt)=-(kz)/(20+2t)

Question

A chemical is diluted out of a tank by pumping pure water into the tank and pumping the existing solution out of it, so the volume at any time  t is  20+2t. The amount  z of chemical in the tank decreases at a rate proportional to  z and inversely proportional to the volume of solution in the tank. Which equation describes this relationship? Choose 1 answer: (A)  (dz)/(dt)=-(k(20+2t))/(z) (B)  (dz)/(dt)=kz-(1)/(20+2t) (C)  (dz)/(dt)=k(20+2t)-(1)/(z) (D)  (dz)/(dt)=-(kz)/(20+2t)

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Identify Problem Statement: The problem states that the rate of decrease of the chemical $ z $ is proportional to $ z $ and inversely proportional to the volume of the solution in the tank, which is $ 20 + 2t $. This means that the rate of change of $ z $ with respect to time $ t $, denoted as $ \frac{dz}{dt} $, should be equal to $ -kz $ divided by the volume $ 20 + 2t $, where $ k $ is the proportionality constant. The negative sign indicates that $ z $ is decreasing.Formulate Rate of Change: We can write the relationship as $ \frac{dz}{dt} = -\frac{kz}{20 + 2t} $. This matches one of the given answer choices.Match with Given Options: Comparing the derived equation with the given options, we find that option (D) $ \frac{dz}{dt} = -\frac{kz}{20 + 2t} $ is the correct description of the relationship.

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