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Answer:
1.857 minutes
Step-by-step explanation:
From the given information:
Consider the amount of salt present in the tank as x(t) at a given time (t); &
The volume of the solution = V(t)
At t = 0 i.e. (at initial conditions) x(0) = 30; and V(0) = 50
However, the overall increase taken place for 1 gallon per minute is:
V(t) = 50 + t
The amount of salt x(t) at any given point for time (t) is;
After 5 gallons of solution exit per minute; the concentration of the salt solution changes at:
Taking the integral of what we have above, we get:
In x(t) = - 5 In(t + 50) + In (C)
In x(t) = In (t+ 50)⁻⁵ + In (C)
In x(t) = In C ( t + 50)⁻⁵
x(t) = C(t + 50)⁻⁵ (General solution)
To estimate the required solution; we apply the initial conditions x(0) = 30;
Thus;
x(0) = C(50)⁻⁵ = 30
⇒ C = 30 × 50⁵
Hence; x(t) = 30 × 50⁵ × (t + 50)⁻⁵
The above expression can be re-written as:
i.e.
50 = 0.964192504t + 48.2096252
50 - 48.2096252 = 0.964192504t
1.7903748 = 0.964192504t
t = 1.7903748 / 0.964192504
t ≅ 1.857 minutes
We can thereby conclude that the estimated time it will require until there are 25 pounds of salt in the tank is 1.857 minutes.