Answer: The amount of salt in the tank after 8 minutes is 36.52 pounds.
Step-by-step explanation:
Salt in the tank is modelled by the Principle of Mass Conservation, which states:
(Salt mass rate per unit time to the tank) - (Salt mass per unit time from the tank) = (Salt accumulation rate of the tank)
Flow is measured as the product of salt concentration and flow. A well stirred mixture means that salt concentrations within tank and in the output mass flow are the same. Inflow salt concentration remains constant. Hence:
By expanding the previous equation:
The tank capacity and capacity rate of change given in gallons and gallons per minute are, respectivelly:
Since there is no accumulation within the tank, expression is simplified to this:
By rearranging the expression, it is noticed the presence of a First-Order Non-Homogeneous Linear Ordinary Differential Equation:
, where .
The solution of this equation is:
The salt concentration after 8 minutes is:
The instantaneous amount of salt in the tank is:
Since 7 is greater than 5, the 9 gets rounded. It would be 12.0.
Answer:
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Going off the idea that this is 4/(y+2) - 9/(y-2) = 9/(y^2-4), let's first rewrite y^2-4 in the form of a^2-b^2, where a=y and b=2.
y/(y+2)-9/(y-2)=9/(y^2-2^2)
Now, use difference of squares: a^2-b^2=(a+b)(a-b)
4/(y+2)-9/(y-2)=9/((y+2)(y-2))
Multiply both sides by the Least Common Denominator: (y+2)(y-2)
4(y-2)-9(y+2)=9
Then keep on simplifying until you get your answer
-5y-26=9
-5y=9+26
-5y=35
y=35/-5
y=-7
Yay! Our answer is y=-7!