Question

There is a tank with 100L of water where 4kg of salt is dissolved. You open a faucet to add a salt solution of .6kg/L at the constant speed of 10 L/min. When do you have to close the faucet if you want the concentration of the salt solution in the tank to be .25kg/L in the tank? Find the time it takes after the faucet is open to the nearest minute.

Answer 1

Let's start by calculating the initial concentration of salt in the tank:

4 kg of salt is dissolved in 100 L of water, so the initial concentration of salt in the tank is:

4 kg / 100 L = 0.04 kg/L

We want to increase the concentration of salt in the tank to 0.25 kg/L by adding a salt solution of 0.6 kg/L at a constant rate of 10 L/min.

Let's assume that t is the time in minutes that the faucet has been open. During this time, the volume of water that has been added to the tank is 10t liters.

The amount of salt that has been added to the tank during this time is:

0.6 kg/L x 10 L/min x t min = 6t kg

The total amount of salt in the tank after t minutes is:

4 kg + 6t kg

The total volume of water in the tank after t minutes is:

100 L + 10t L

The concentration of salt in the tank after t minutes is:

(4 kg + 6t kg) / (100 L + 10t L)

We want this concentration to be 0.25 kg/L, so we can set up the following equation:

(4 kg + 6t kg) / (100 L + 10t L) = 0.25 kg/L

Simplifying this equation, we get:

16 kg + 24t kg = 25 L + 2.5t L

21.5t = 9 L

t = 9 L / 21.5 = 0.42 hours = 25.2 minutes (rounded to the nearest minute)

Therefore, you need to close the faucet after approximately 25 minutes to achieve a concentration of 0.25 kg/L in the tank.