Problem 54 - Algebra 2

Question

WATER TREATMENT A certain water filtration system can remove 80\(\%\) of the contaminants each time a sample of water is passed through it. If the same water is passed through the system three, what percent of the original contaminants will be removed from the water sample?

Step-by-Step Solution

Answer

99.2% of the original contaminants are removed after three passes.

Step 1: Determine Filtration Per Pass

Each time the water passes through the system, 80% of the contaminants are removed. This means that 20% of the contaminants remain. Thus, we can say that after a single pass through the filter, the water retains 20% of its contaminants.

Step 2: Calculate Remaining Contaminants After Three Passes

We need to calculate the percentage of contaminants remaining after three passes. With each pass retaining 20% of the contaminants, we can use this formula for each pass:\[ R_n = R_0 imes 0.2^n \]Where \(R_n\) is the percentage of contaminants remaining after \(n\) passes, and \(R_0\) is the initial 100%. After three passes:\[ R_3 = 100 imes 0.2^3 = 100 imes 0.008 = 0.8 \]

Step 3: Calculate Total Percent of Contaminants Removed

Initially, there were 100% of the contaminants in the water. After three filtration passes, 0.8% of the original contaminants remain. Thus, the percentage of contaminants removed is:\[ 100 - 0.8 = 99.2 \]

Step 4: Final Answer Interpretation

The calculation shows that 99.2% of the original contaminants are removed after three passes through the water filtration system.

Key Concepts

Percentage Problems

Solving percentage problems involves understanding how a part relates to a whole, usually expressed as a fraction or a percentage. In our water filtration exercise, the filtration system removes a certain percentage of contaminants. Here, 80% of contaminants are removed each time water passes through the filter, implying that 20% of the contaminants remain.

To find out how much is removed after multiple passes, you can use the concept of exponential decay. Each subsequent passage through the filter further reduces the number of contaminants by another 80%. Calculating this requires a basic understanding of percentage and exponential operations.

• Initially, you start with 100% of the contaminants.
• After the first pass, 80% is removed, and only 20% remains.
• The task is to find how much is removed after repeated filtration.

Understanding these percentages is crucial because they tell you how efficient the filtration process is over multiple cycles.

Filtration Process

A filtration process is a method used to separate impurities or contaminants from water by passing it through a filter. The effectiveness of a filtration system can be measured by its ability to remove a specific percentage of unwanted materials.

In our example, the water filtration system is designed to remove 80% of contaminants each time it is used. This means a large portion of contaminants is taken out with every pass through the system. The idea is similar to repeatedly cleaning a piece of fabric; each time you wash it, it becomes cleaner.

The system we're examining reduces contaminants exponentially due to repetitive application. This exponentiation is crucial to understanding how even a small increase in the number of times the filter is used can lead to a significant reduction in contaminants. That's why the process is calculated step-by-step, reflecting how each passage through the filter affects the outcome.

Contaminant Removal

Contaminant removal is an essential aspect of water treatment. It refers to the reduction of impurities or undesirable materials from the water supply to acceptable levels.

In this problem, each filtration pass removes 80% of contaminants. Over multiple passes, the total amount of contaminants significantly decreases due to exponential decay. This is evident in the calculation where, after three passes, 99.2% of the original contaminants are removed.

The key to understanding contaminant removal here is mastering the exponential reduction. With each pass, an additional 80% of the remaining contaminants are removed, leaving behind only a small fraction.

• The initial pass drastically reduces contaminants.
• Subsequent passes further cut down the number of unwanted substances.

By the time three passes are completed, only a tiny 0.8% of the original contaminants linger in the water. This highlights the filtration system's effectiveness in purifying water by removing almost all unwanted material.