Chapter 13.4, Problem 35E

Salt Concentration

1. (a) A tank contains 5000 L of pure water. Brine that contains 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 g/L) is $C(t)=\frac{30t}{200+t}$
2. (b) What happens to the concentration as t → ∞?

To determine

To Show: The concentration of salt after t minutes is $C\left(t\right)=\frac{30t}{200+t}$.

Explanation of Solution

Proof:

The initial quantity of pure water in the tank is $5000\text{L}$ and then tank is poured with $25\text{L}/\text{min}$ of brine.

After time t the quantity of water in the tank is,

$5000+25t$

Initially, there is no salt in the tank, brine contains $30\text{g}/\text{L}$ of salt in it. So, amount of salt after time t in the tank is $\left(25t\right)30$.

Salt concentration after time t is amount of salt after time t divided by total amount of water in the tank.

Mathematically,

$\begin{array}{c}C\left(t\right)=\frac{\left(25t\right)30}{5000+25t}\\ =\frac{25\left(30t\right)}{25\left(200+t\right)}\\ =\frac{30t}{200+t}\end{array}$

Hence, the concentration of salt after time t is $\frac{30t}{200+t}$.

To determine

To Find: The concentration of salt when $t\to \infty$.

Answer to Problem 35E

The concentration of salt when $t\to \infty$ is $30\text{g}/\text{L}$.

Explanation of Solution

Given:

The concentration of salt as a function of time t is $C\left(t\right)=\frac{30t}{200+t}$.

Here,

• $C\left(t\right)$ is concentration of salt in $\text{g}/\text{L}$.
• t is time in minutes.

Calculation:

When $t\to \infty$ concentration of salt is given by,

$\underset{t\to \infty }{\lim }C\left(t\right)=\underset{t\to \infty }{\lim }\frac{30t}{200+t}$

Divide numerator and denominator by t and compute the limit.

$\begin{array}{c}\underset{t\to \infty }{\lim }C\left(t\right)=\underset{t\to \infty }{\lim }\frac{\left(\frac{1}{t}\right)30t}{\left(\frac{1}{t}\right)\left(200+t\right)}\\ =\underset{t\to \infty }{\lim }\frac{30}{\left(\frac{200}{t}+1\right)}\\ =30\left[\because \underset{t\to \infty }{\lim }\frac{200}{t}=0\right]\end{array}$

Thus, the concentration of salt when $t\to \infty$ is $30\text{g}/\text{L}$.