Problem 2 - Active Calculus

Question

A tank contains \(50 \mathrm{~kg}\) of salt and \(1000 \mathrm{~L}\) of water. A solution of a concentration \(0.025 \mathrm{~kg}\) of salt per liter enters a tank at the rate \(10 \mathrm{~L} / \mathrm{min}\). The solution is mixed and drains from the tank at the same rate. (a) What is the concentration of our solution in the tank initially?

Step-by-Step Solution

Answer

The initial concentration is 0.05 kg/L.

Step 1: Define initial conditions

Initially, the tank contains 50 kg of salt and 1000 L of water.

Step 2: Calculate initial concentration

The concentration of the solution is given by the amount of salt divided by the volume of water. Thus, the initial concentration can be calculated as follows: \[ \text{Initial concentration} = \frac{50 \text{ kg}}{1000 \text{ L}} = 0.05 \text{ kg/L} \]

Step 3: Verify the calculation

Double-check the arithmetic to ensure that the concentration is correct: \[ 50 \text{ kg} \times \frac{1}{1000 \text{ L}} = 0.05 \text{ kg/L} \]

Key Concepts

Initial Concentration

Let's start by understanding the initial concentration in the tank. Initially, the tank contains 50 kg of salt and 1000 L of water.
To find the concentration, you'll need to divide the amount of salt by the volume of water. This gives the concentration of salt per liter in the solution. So, using the formula: \[ \text{Initial concentration} = \frac{50 \text{ kg}}{1000 \text{ L}} = 0.05 \text{ kg/L} \] This means that for every liter of water in the tank, there are 0.05 kg of salt.

Rate of Flow

The rate of flow is crucial for understanding how the salt concentration changes over time.
In this problem, a solution containing 0.025 kg of salt per liter is entering the tank at 10 L/min. Simultaneously, the mixed solution drains from the tank at the same rate of 10 L/min.
This balance in the rate of inflow and outflow ensures that the volume of liquid in the tank remains constant at 1000 L.
Since both the inflow and outflow rates are the same, the concentration change over time will solely depend on the mixing process inside the tank.

Solution Mixing

Mixing is vital in distributing the salt evenly in the water.
In this problem, the incoming solution mixes thoroughly with the tank's existing content.
This ensures that the outgoing solution has the same concentration as the solution inside the tank. The mixing process happens continuously, maintaining a uniform salt concentration throughout the liquid.
This concept is particularly important because it affects the salt's concentration variations over time.

Arithmetic Verification

Verifying your calculations is essential to avoid errors.
In our scenario, we double-checked the initial concentration calculation to ensure accuracy: \[ 50 \text{ kg} \times \frac{1}{1000 \text{ L}} = 0.05 \text{ kg/L} \]
This reassures us that the initial concentration of 0.05 kg/L is correct.
Arithmetic verification helps confirm that all calculations leading up to your final answer are accurate and reliable.

Calculation of Concentration

Finally, the concentration of the solution in the tank can be calculated and verified.
The initial concentration calculation already showed us that the starting point is 0.05 kg/L.
To determine how the concentration changes over time, you need to account for the inflow and outflow dynamics. Using differential equations, you can predict how the concentration will adjust given the steady rate of 10 L/min and the initial conditions.
This step is fundamental in understanding the overall change in concentration in any given period.