Problem 58 - Precalculus Mathematics for Calculus

Question

A health clinic uses a solution of bleach to sterilize Petri dishes in which cultures are grown. The sterilization tank contains 100 gals 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 about 3.06 gallons with pure bleach.

Step 1: Understand the Initial Setup

The tank initially contains 100 gallons of a 2% bleach solution. This means 98% is water and 2% is bleach. We need to increase this to a 5% solution.

Step 2: Calculate Initial Amount of Bleach

The initial amount of bleach in the solution is given by \(0.02 \times 100 = 2\) gallons.

Step 3: Set Up the Equation for Final Concentration

Let \(x\) be the number of gallons of the original solution to be drained and replaced with pure bleach. After draining and adding, the solution will still be 100 gallons, but with greater bleach concentration.

Step 4: Expression for Bleach Content After Replacement

After draining, \(100 - x\) gallons remain, containing \(2\%\) bleach (i.e., \(0.02(100 - x)\) gallons of bleach). Adding \(x\) gallons of pure bleach (100% bleach) results in a total of \(0.02(100-x) + x\) gallons of bleach.

Step 5: Final Bleach Concentration Requirement

For the final solution, the bleach needs to be 5% of the 100 gallons, which is 5 gallons. Set up the equation: \[ 0.02(100 - x) + x = 5 \]

Step 6: Solve the Equation

Solve for \(x\) in the equation: \[ 0.02(100 - x) + x = 5 \]First, expand: \(2 - 0.02x + x = 5\)Combine the terms of \(x\): \(2 + 0.98x = 5\)Subtract 2: \(0.98x = 3\)Solve for \(x\): \(x = \frac{3}{0.98} \approx 3.06\) gallons.

Step 7: Conclusion

Approximately 3.06 gallons of the 2% solution need to be drained and replaced with pure bleach.

Key Concepts

Percentage Calculation

To tackle problems like this, understanding percentage calculation is essential. When we deal with percentages, we're talking about parts of 100. In this problem, we start with a 2% bleach solution in 100 gallons of liquid. This means that 2% of this volume is bleach, and we need this interpretation to figure out how much bleach is initially present.

To find how much bleach is in the initial solution, multiply the total volume (100 gallons) by the bleach percentage as a decimal, which in this case is 0.02. So, the calculation is:

• Initial bleach content = 100 gallons × 0.02 = 2 gallons

Knowing how to convert percentages to decimals and use them to find portions of a whole is a fundamental skill in mixture problems. You often start with a percentage, then convert it to a decimal for multiplication.

Concentration

Concentration refers to how much of a substance (like bleach) is present in a mixture compared to the total quantity of the mixture. It can be expressed in percentages, making it easier to visualize part-to-whole relationships.

In this exercise, the goal is to increase the concentration of bleach in the solution from 2% to 5%. This can be understood by thinking about needing more bleach compared to the water in the mixture. To change the concentration effectively, manipulation through drainage and replacement is applied.

The existing solution contains 98% water and 2% bleach. To increase the bleach to the desired 5%, we need to first eliminate some of the lower concentration solution and replace it with pure bleach (which is 100% concentration).

Thus, thinking critically about concentration allows you to manipulate the amounts to match desired levels while still keeping the total mixture at 100 gallons.

Equation Solving

Equation solving is crucial in converting real-world problems into solvable mathematical representations. Once we know we want a final concentration of 5% bleach, setting up an equation helps us determine how much solution to replace.

We seek the value of \( x \), which represents the gallons of the original solution to be replaced. The equation derived is:

• \(0.02(100 - x) + x = 5\)

This equation represents the final situation after replacement. By simplifying:

• First, distribute: \(2 - 0.02x + x = 5\)
• Combine like terms: \(2 + 0.98x = 5\)
• Subtract 2 from both sides: \(0.98x = 3\)
• Divide by 0.98 to solve for \(x\): \(x = \frac{3}{0.98} \approx 3.06\)

This manipulation shows how carefully setting up and solving equations can find how much of the solution to alter. It ties together understanding percentages and concentrations into a concrete mathematical action plan.