Question

A tank consists of 50 liters of fresh water. Two liters of brine each litre containing 5 gms of dissolved salt are running into tank per minute ; the mixture is kept uniform by stirring , and runs out at the rate of one litre per minute.If $m$ grams of salt are present in the tank after t minute. expression of $m$ in terms of $t$ be$m=kt(r+\frac{2p+t}{p+t})gm$. Find $\left(p/k\right)+r$ ?

Answer 1

Input rate = 10 gm/min
After time t volume of tank = 50 + (2-1)t
Concentration of salt in time $t=\frac{m}{50+t}gm/lit$
Output rate $=\frac{m}{50+t}.1gm/min$
$\Rightarrow \frac{dm}{dt}=\frac{-m}{50+t}+10$

$\Rightarrow m(50+t)=\int 10(50+t)dt$

$\Rightarrow m(50+t)=500t+5t^2+c$
$\Rightarrow (\because att=0,m=0\Rightarrow c=0)$
$\Rightarrow m(50+t)=5(100t+t^2)$
$\Rightarrow m=5t\left(\frac{100+t}{50+t}\right)gm$