Problem 38 - 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 of the solution with pure bleach.

Step 1: Define the variables

Let \( x \) be the number of gallons of the solution to be drained and replaced with pure bleach. We need to find the value of \( x \).

Step 2: Understand the initial concentration

There are initially 100 gallons of a 2\( \% \) bleach solution. This means there are \( 0.02 \times 100 = 2 \) gallons of bleach in the solution.

Step 3: Calculate the final amount of bleach needed

To have a 5\( \% \) bleach solution in the 100 gallons mixture, we need \( 0.05 \times 100 = 5 \) gallons of bleach.

Step 4: Set up an equation for the new concentration

When \( x \) gallons of solution are drained and replaced by \( x \) gallons of pure bleach, the amount of bleach in the solution will be \( 2 + x \). We need this to equal the final bleach content required. Therefore, the equation is: \( 2 + x = 5 \)

Step 5: Solve the equation

Solve the equation \( 2 + x = 5 \) to find \( x \).\[x = 5 - 2 = 3\] So, \( x = 3 \) gallons.

Key Concepts

Bleach Concentration

Bleach concentration refers to the amount of bleach present within a solution. It is important because it determines the solution’s effectiveness in tasks like sterilization. The percentage concentration indicates how much of the total solution is made up of bleach. In this problem, the health clinic initially uses a 2\(\%\) bleach solution. This means that in every 100 gallons of their solution, 2 gallons are bleach. Understanding bleach concentration helps in determining how much more or less bleach is required to achieve the desired effectiveness in cleaning or sterilizing. Keeping the right concentration is crucial for safety and effectiveness without causing damage.

Solution Mixing

Solution mixing involves combining different substances to create a solution with desired properties. In this case, the problem involves adjusting a bleach solution to ensure safe and effective sterilization of petri dishes. The original solution is a combination of household bleach and water. To increase the concentration of bleach, one can either increase the amount of bleach or decrease the amount of water. However, in this scenario, replacing some of the lower concentration solution with pure bleach is chosen to achieve the needed 5\(\%\) concentration. Successfully mixing solutions requires a clear understanding of the proportions and the desired result in terms of bleach concentration.

Percentage Concentration

Percentage concentration, often expressed as a percentage, measures the proportion of a particular component in a total mixture. For the health clinic's bleach solution, percentage concentration is used to measure how much of the total solution consists of bleach. Originally, the clinics use a 2\(\%\) solution, meaning only 2 gallons out of every 100 are bleach. The exercise aims to increase this concentration to 5\(\%\).

• To do this, you need to compute how much pure bleach should be added.
• This involves understanding how much bleach would constitute 5\(\%\) in the existing 100 gallons.
• Calculating adequately ensures safety and desired results.

Adjusting percentages meticulously is crucial as it ensures that the mixture will perform its intended function effectively.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. In mixture problems like this one, equations help determine how much of a component needs to be added or removed to achieve a desired concentration. Here, introducing the variable \(x\) simplifies the complexity: it represents the number of gallons to drain and replace with bleach.

• The equation \(2 + x = 5\) represents the current bleach quantity plus the replaced bleach equals the needed bleach quantity.
• The solution to this equation, \(x = 3\), tells us that 3 gallons need to be replaced.
• Using algebraic equations allows precise adjustments to solution concentrations.

These make it possible to plan effectively and ensure the desired mix's safety and efficacy. Algebra brings clarity and precision in solving such real-life problems.