Question

A tank contains $40l$ of solution containing $2g$ of substance per liter. Salt water containing $3g$ of this substance per liter runs in at the rate of $4l/min$ and the well-stirred mixture runs out at the same rate. Find the amount of substance in the tank after $15minutes$ (in grams).

• A. $110.075$
• B. $121.075$
• C. $111.075$
• D. $211.075$

Answer 1

The correct option is C $111.075$

Initially there is $40\times 2=80$ grams

Let $Q$ be the amount of substance in the tank and $\frac{dQ}{dt}$ be the rate of change of amount in the tank

We get the equation as $\frac{dQ}{dt}+\frac{1}{10}Q=12$

It is linear equation and the solution of it is $Q=120+C{e}^{-\frac{t}{10}}$

At $t=0$ , the value of $Q$ is $80$ grams

$\Rightarrow C=-40$

Therefore the equation is $Q=120-40e^{-\frac{t}{10}}$

At $t=15$ , we get $Q=120-40e^{-\frac{15}{10}}=111.075$