A 200-gallon tank is currently half full of water that contains 50 pounds of salt. A solution containing 1 pounds of salt per gallon enters the tank at a rate of 6 gallons per minute, and the well-stirred mixture is withdrawn from the tank at a rate of 6 gallons per minute. How many pounds of salt are in the tank 10 minutes later? Round your answer to 2 decimal places.
I got 104.82 but it is wrong.
Please explain the answer and how to get it.
Why don't you explain your answer, and we can see whether your math is sound.
Did you start (with s(t) being the amount of salt present)
ds/dt = 1 - 6/100 s
s(0) = 50
?
Surely you can see that 104.82 is way off. Even with no salt draining out, only 110 lb would be present after 10 minutes!
my equation was y=900+C/e^(1/150)t
yeah, but how did you get it?
Where did the 900 and 150 come from?
With only 50 lb of salt at t=0, I don't see how it works. What is y measuring?
I started out by reasoning that with no draining,
ds/dt = 6
since since each minute 6 lbs of salt are added. OUCH! Typo, since I said 1 (because of 1 lb/gal, but I ignored the fact that 6 gal/min are added)
But, at the same time 6 of the 100 gallons drain out, taking 6/100 of the salt present with them. Hence the differential equation.
To solve this problem, we need to track the amount of salt entering and leaving the tank over a period of time.
Let's break down the problem step by step:
1. Determine the initial amount of salt in the tank:
- We are given that the tank is currently half full, which means it contains 100 gallons of water.
- Since the solution entering the tank has a concentration of 1 pound of salt per gallon, the initial amount of salt in the tank is 100 pounds (100 gallons * 1 pound/gallon).
2. Calculate the amount of salt entering the tank per minute:
- The rate of salt entering the tank is given as 6 gallons per minute, with a concentration of 1 pound of salt per gallon.
- Therefore, the amount of salt entering the tank per minute is 6 pounds (6 gallons * 1 pound/gallon).
3. Calculate the amount of salt leaving the tank per minute:
- The rate of withdrawal is also given as 6 gallons per minute, but we need to determine the concentration of salt in the mixture being withdrawn.
- Since the tank was initially half full and now contains 100 gallons of water, the concentration of salt in the tank is 50 pounds / 100 gallons = 0.5 pound/gallon.
- Therefore, the amount of salt leaving the tank per minute is 6 gallons * 0.5 pound/gallon = 3 pounds.
4. Calculate the amount of salt in the tank after 10 minutes:
- Over a span of 10 minutes, the amount of salt entering and leaving the tank is equal, as both rates are 6 gallons per minute.
- The total amount of salt entering and leaving the tank in 10 minutes is (6 pounds/minute - 3 pounds/minute) * 10 minutes = 30 pounds.
- The updated amount of salt in the tank after 10 minutes is then 100 pounds + 30 pounds = 130 pounds.
Therefore, the correct answer is 130.00 pounds of salt in the tank 10 minutes later.