Question

(a) A tank contains 5000 L of pure water. Brine that contain\(C\left( t \right) = \frac{{{\bf{30}}t}}{{{\bf{200}} + t}}\)s 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after t minutes (in grams per liter) is

(b) What happens to the concentration as \(t \to \infty \)?

Short answer

It is proved that the concentration is\(C\left( t \right) = \frac{{30t}}{{200 + t}}\).

If\(t \to \infty \) then the concentration of salt in the tank will reach to \(30\;{\rm{g/L}}\).

Step-by-step solution

Step 1:Find the expression for concentration of salt

After t minutes, \(25t\) liters of brine with 30 g of salt is pumped in tank, so the tank has \(\left( {5000 + 2t} \right)\) liters water.

The amount of slat in the tank at this instant is:

\(25t \cdot 30 = 750t\)

The salt concentration in the tank at time t can be calculated as is given by the expression:

\(\begin{aligned}C\left( t \right) &= \frac{{750t}}{{5000 + 25t}}\\ &= \frac{{30t}}{{200 + t}}\;{\rm{g/L}}\end{aligned}\)

Find the value of concentration of salt as

\(t \to \infty \)

Find the value of \(C\left( t \right)\) as \(t \to \infty \).

\(\begin{aligned}\mathop {\lim }\limits_{t \to \infty } C\left( t \right) &= \mathop {\lim }\limits_{t \to \infty } \frac{{30t}}{{200 + t}}\\ &= \mathop {\lim }\limits_{t \to \infty } \frac{{\frac{{30t}}{t}}}{{\frac{{200 + t}}{t}}}\\ &= \mathop {\lim }\limits_{t \to \infty } \frac{{30}}{{\frac{{200}}{t} + 1}}\\ &= \frac{{30}}{{0 + 1}}\\ &= 30\end{aligned}\)

So, as \(t \to \infty \) the concentration of salt in the tank will reach to \(30\;{\rm{g/L}}\).