Concentration of salt=1.5/10=0.15kg/L. Tank is losing 0.15kg/min.
3L contains 0.45kg of salt. Tank is losing 1L of solution per minute.
Tank gains 0.5kg/L×2=1kg/min, but is losing 0.15kg/min.
(If the tank is filling at the rate of v L/min, the salt is increasing by 0.5v kg/min.)
Net gain is 1-0.15=0.85kg/min=0.85t kg over t mins. (Net gain of salt is 0.5v-0.15 kg/min at fill rate of v L/min.)
Tank is gaining 2L of solution per minute. Net gain of solution is 1L/min. (If fill rate is v, net gain of solution is v-1 L/min.)
(a) Amount of salt in the water after t mins=initial amount+0.85t.
That is, 0.45+0.85t kg.
(b) The tank has to gain 7L before it overflows. That’s 7 minutes.
When t=7, the amount of salt=0.45+0.85×7=0.45+5.95=6.4kg.
So there would be 10L of water containing 6.4kg of salt, so the concentration would be 6.4/10=0.64kg/L. Therefore the concentration does reach 0.45kg/L before the tank overflows.
(c) If the tank is filling at an overall rate of v-1 L/min, it would take t=7/(v-1) minutes to reach 10L capacity after starting with 3L. Therefore, the concentration of salt would be (0.45+(0.5v-0.15)(7/(v-1))/10.
In words this means we use the formula from (a) to find the amount of salt in solution after time t=7/(v-1) (overall fill rate is v-1 L/min). We are still losing salt at the rate of 0.15kg/min but we are gaining 0.5v kg/min, that’s a net gain of 0.5v-0.15 kg/min. When the tank is full this amount of salt will be in solution, so to find the concentration we need to divide the amount by the capacity of 10L.
Now to solve for v:
(0.45+(0.5v-0.15)(7/(v-1)))/10=0.45,
0.45+(0.5v-0.15)(7/(v-1))=4.5 kg,
(0.5v-0.15)(7/(v-1))=4.5-0.45=4.05 kg (top-up),
(3.5v-1.05)/(v-1)=4.05,
3.5v-1.05=4.05(v-1)=4.05v-4.05,
4.05-1.05=(4.05-3.5)v,
3=0.55v, v=3/0.55=5.45L/min (60/11 L/min).
CHECK
Fill rate=60/11 L/min of solution containing 30/11 kg/min of salt.
Tank is still losing 0.15kg/min through leaks. Net gain=30/11-0.15 kg/min.
Tank had 3L of solution containing 0.45kg of salt before filling started.
Another 7L of solution is needed to top up the tank. The tank is filling at the rate of 60/11 L/min but emptying at the rate of 1L/min. Overall fill rate is 60/11-1=49/11 L/min. So it will take 7/(49/11)=11/7 mins to reach 10L capacity. The amount of salt will be
0.45+(30/11-0.15)(11/7)=
0.45+(30-1.65)/7=
0.45+4.05=4.5kg. This is in 10L of solution, 0.45kg/L.
This confirms the fill rate to be 60/11 L/min.
