Tank A contains 80 gallons of water in which 20 pounds of salt has been dissolved. Tank B contains 30 gallons of water in which 5 pounds of salt has been dissolved. A brine mixture with a concentration of 0.5 pounds of salt per gallon of water is pumped into tank A at the rate of 4 gallons per minute. The well-mixed solution is then pumped from tank A to tank B at the rate of 6 gallons per minute. The solution from tank B is also pumped through another pipe into tank A at the rate of 2 gallons per minute, and the solution from tank B is also pumped out of the system at the rate of 4 gallons per minute. The correct differential equations with initial conditions for the amounts, \( x(t) \) and \( y(t) \), of salt in tanks A and B, respectively, at time \( t \) are Select the correct answer. \( \frac{d x}{d t}=2-\frac{x}{40}+\frac{y}{5}, \frac{d y}{d t}=\frac{x}{40}-\frac{y}{3}, x(0)=20, y(0)=5 \) \( \frac{d x}{d t}=2-\frac{3 x}{40}+\frac{y}{15}, \frac{d y}{d t}=\frac{3 x}{40}-\frac{y}{5}, x(0)=20, y(0)=5 \) \( \frac{d x}{d t}=4-\frac{3 x}{40}+\frac{y}{15}, \frac{d y}{d t}=\frac{3 x}{40}-\frac{y}{5}, x(0)=20, y(0)=5 \) \( \frac{d x}{d t}=4-\frac{x}{40}+\frac{y}{5}, \frac{d y}{d t}=\frac{x}{40}-\frac{y}{3}, x(0)=20, y(0)=5 \) \( \frac{d x}{d t}=2-\frac{3 x}{40}+\frac{y}{5}, \frac{d y}{d t}=\frac{x}{40}-\frac{y}{5}, x(0)=20, y(0)=5 \) \( x \)
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The Deep Dive
Did you know that the concept of differential equations dates back to the 18th century, when mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz were looking to describe change in dynamic systems? They laid the groundwork for what we now use in everything from physics to economics! Fast forward, these equations enable us to model situations like the mixing of saltwater in tanks, just like in your problem. Solving differential equations can seem daunting, but a few strategies can help. Start by clearly defining your variables and rates of change, just like you've done. A common mistake is to forget to account for the changing concentrations as solutions are mixed. It’s like making a delicious sauce—you need to balance the ingredients! Remember, mixing things up can lead to unexpected flavors, or in this case, unexpected salt concentrations!
