Problem 50 - Algebra and Trigonometry

Question

Mixture Problem A health clinic uses a solution of bleach to sterilize petri dishes in which cultures are grown. The sterilization tank contains 100 gal of a solution of 2\(\%\) ordinary household bleach mixed with pure distilled water. New research indicates that the concentration of bleach should be 5\(\%\) for complete sterilization. How much of the solution should be drained and replaced with bleach to increase the bleach content to the recommended level?

Step-by-Step Solution

Answer

Drain and replace 3 gallons for a 5% bleach solution.

Step 1: Identify Initial Conditions

We have a 100-gallon tank containing a 2% bleach solution. This means there are initially 2 gallons of bleach and 98 gallons of water.

Step 2: Set Up the Mixture Equation

Let \( x \) be the amount of the solution to be drained and replaced with pure bleach. The remaining solution then contains \( 100 - x \) gallons, which will still be at 2% bleach concentration.

Step 3: Calculate Bleach After Drainage

After removing \( x \) gallons of the mixture and replacing it with pure bleach, the amount of bleach in the tank (in gallons) will be: \[ 0.02(100 - x) + x = 5 \text{ gallons of bleach.} \]

Step 4: Formulate the Equation

We know that the initial bleach plus the new bleach should equal the desired concentration: \[ 0.02(100) + x = 5. \] Calculate \( x \) to find the amount of pure bleach needed.

Step 5: Solve for \(x\)

Simplify the equation: \[ 2 + x = 5. \] Subtract 2 from both sides to isolate \( x \): \[ x = 5 - 2 = 3. \]

Step 6: Verify the Solution

Substitute \( x = 3 \) back into the scenario: We initially have 2 gallons of bleach and we add 3 gallons of pure bleach. This results in 5 gallons of bleach in 100 gallons of solution, which gives us a 5% bleach concentration.

Key Concepts

Sterilization Solution

Sterilization solutions are crucial in maintaining hygiene and preventing contamination, especially in a health clinic. They ensure that all microorganisms, including bacteria and fungi, are effectively eliminated from surfaces and tools. In the context of our problem, a sterilization solution consists of a mixture of household bleach, which is an affordable and effective disinfectant. Household bleach is typically made up of sodium hypochlorite and is used widely for its antimicrobial properties.

When we discuss a sterilization solution, it is important to remember the role of concentration in its effectiveness. In this exercise, the original solution has a concentration of 2\(\%\) bleach, which needs to be adjusted to 5\(\%\) for optimal sterilization. This adjustment ensures that all surfaces cleaned with this solution are free from harmful pathogens. Mixing the correct concentration is vital for both ensuring safety and avoiding damage from over-concentration.

Bleach Concentration

Bleach concentration refers to the amount of bleach present in a mixture relative to the total volume of the solution. It is usually expressed as a percentage. In our example, the starting concentration was 2\(\%\), meaning 2 gallons of bleach in the 100-gallon solution. However, to reach the required sterilization effectiveness, the concentration needs to be elevated to 5\(\%\).

The concentration is key to the efficacy of the sterilization process; a lower concentration might not kill all pathogens, while too high a concentration could potentially damage surfaces or equipment. Adjusting the concentration involves algebraic calculations to determine how much of the original mix to replace with pure bleach. This careful balance is essential for achieving a solution that is both effective and safe to use. Proper concentration ensures the bleach works well without causing adverse side effects.

Algebraic Equation

Algebraic equations play a significant role in solving mixture problems. They allow us to set up a mathematical representation of the problem, making it easier to figure out the solution. In this problem, we used an algebraic equation to understand how much initial solution should be removed and replaced with pure bleach to reach the desired 5\(\%\) concentration.

We set the equation using \( x \), which stands for the amount of solution to be drained. Additionally, the components of the current concentration and the goal concentration are expressed in terms of amounts and percentages. Solving an algebraic equation involves simplifying and rearranging terms to solve for the unknown variable. In this case, solving the equation \( 2 + x = 5 \) for \( x \) gave us the amount of bleach required to achieve the desired concentration. This approach illustrates the practical utility of algebra in problem-solving.

Mixture Equation

A mixture equation is essentially a type of algebraic equation specifically used to solve mixture problems involving different concentrations. In the given exercise, the mixture equation helps us determine how much solution to replace to achieve the desired bleach concentration.

The mixture equation set up from the exercise is: \( 0.02(100) + x = 5 \). This represents the existing amount of bleach and adds \( x \), the amount of pure bleach added. This equation balances both the initial and the changed conditions of the solution.

Mixture equations are useful because they structure the problem logically, illustrating the relationship between different elements like volume and concentration. They provide a clear path from given data to the solution, making complex problems with several variables much more manageable. Understanding how to set and solve such equations is invaluable in chemistry, engineering, and various applied sciences.