Mathematics

Mathematics

Problem

A tank initially contains 200 L of brine with 30 kg of dissolved salt. Brine with a concentration of 0.10 kg/L flows in at 8 L/min, while the mixture leaves at 5 L/min. The volume therefore increases by 3 L/min. Let Q(t)Q(t) be the amount of salt (in kg) in the tank tt minutes after the process begins. Which of the following expressions gives Q(t)Q(t)? a) Q(t)=0.1(200+3t)+10(200200+3t)2/3Q(t)=0.1(200+3t)+10\left(\dfrac{200}{200+3t}\right)^{2/3} b) Q(t)=0.1(200+3t)+10(200200+3t)5/3Q(t)=0.1(200+3t)+10\left(\dfrac{200}{200+3t}\right)^{5/3} c) Q(t)=0.1(200+3t)+10(200+3t200)5/3Q(t)=0.1(200+3t)+10\left(\dfrac{200+3t}{200}\right)^{5/3} d) Q(t)=0.8(200+3t)+10(200200+3t)5/3Q(t)=0.8(200+3t)+10\left(\dfrac{200}{200+3t}\right)^{5/3}
Intuitive explanation

This is a mixing problem where you need to track how salt quantity changes over time using a rate equation. The key insight is to set up a differential equation by balancing the rate salt enters minus the rate salt leaves—remembering that the concentration of salt leaving depends on the current amount of salt divided by the current volume, which grows as time passes. Once you have dQdt\frac{dQ}{dt}, you solve this first-order linear ODE with the initial condition Q(0)=30Q(0) = 30 kg to find Q(t)Q(t).

Intuitive explanation

This is a mixing problem where you need to track how salt quantity changes over time using a rate equation. The key insight is to set up a differential equation by balancing the rate salt enters minus the rate salt leaves—remembering that the concentration of salt leaving depends on the current amount of salt divided by the current volume, which grows as time passes. Once you have dQdt\frac{dQ}{dt}, you solve this first-order linear ODE with the initial condition Q(0)=30Q(0) = 30 kg to find Q(t)Q(t).

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