Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

Chlorine levels Chlorine is often added to swimming pools to control microorganisms. If the level of chlorine rises above 3 ppm (parts per million), swimmers will experience burning eyes and skin discomfort. If the level drops below \(1 \mathrm{ppm}\), there is a possibility that the water will turn green because of a large algae count. Chlorine must be added to pool water at regular intervals. If no chlorine is added to a pool during a 24 -hour period, approximately \(20 \%\) of the chlorine will dissipate into the atmosphere and \(80 \%\) will remain in the water. (a) Determine a recursive sequence \(a_{n}\) that expresses the amount of chlorine present after \(n\) days if the pool has \(a_{0}\) ppm of chlorine initially and no chlorine is added. (b) If a pool has \(7 \mathrm{ppm}\) of chlorine initially, construct a table to determine the first day on which the chlorine level will drop below \(3 \mathrm{ppm}\).

Step-by-Step Solution

Answer

The chlorine level will drop below 3 ppm on the fourth day.

Step 1: Understand the Chlorine Dissipation Rate

In a 24-hour period, 20% of the chlorine dissipates, leaving 80% remaining for the next day. This means the next day's amount of chlorine is 80% of the previous day's amount.

Step 2: Define the Recursive Formula

The initial amount of chlorine is given as \(a_0\). Each day, chlorine is reduced to 80% of the previous day's amount. Thus, the recursive formula is \(a_{n} = 0.8 \times a_{n-1}\).

Step 3: Identify Initial Conditions

The initial amount of chlorine \(a_0\) is equal to the initial ppm of chlorine in the pool, which is 7 ppm for this exercise.

Step 4: Calculate Chlorine Levels Over Days

To find the chlorine level each day, apply the recursive formula starting from \(a_0 = 7\). Calculate \(a_1, a_2, a_3, ...\) until the level is below 3 ppm:- \(a_1 = 0.8 \times 7 = 5.6\)- \(a_2 = 0.8 \times 5.6 = 4.48\)- \(a_3 = 0.8 \times 4.48 = 3.584\)- \(a_4 = 0.8 \times 3.584 = 2.8672\)

Step 5: Determine the Day When Chlorine Drops Below 3 ppm

From the calculations, on day 4, the chlorine level \(a_4\) drops below 3 ppm.

Key Concepts

Chlorine Dissipation

Chlorine dissipation refers to the natural process where the chlorine in a pool diminishes over time, primarily due to its reaction with contaminants and exposure to sunlight, which can cause it to evaporate. This is why a concrete pool maintenance schedule is important. In the context of our exercise, we're looking at a particular situation where approximately 20% of the chlorine level evaporates every 24 hours.
This means if you start with 100% of chlorine in your pool, only 80% will remain after one day without adding more.
• Day 1: 80% of original chlorine level remains
• Day 2: 80% of Day 1 level = (100% x 0.8) x 0.8 = 64% remains
The percentage of remaining chlorine rapidly decreases with time if no additional chlorine is introduced. It's essential to understand this rate to maintain proper sanitation levels in the pool.

Swimming Pool Maintenance

Proper swimming pool maintenance hinges heavily on regulating the chlorine levels. It's not just about adding chlorine; it involves understanding how much to add and when. Chlorine serves multiple purposes, from killing harmful microorganisms to preventing algae growth, although its rate of dissipation must be factored into regular maintenance activities.
To effectively manage a pool, consider these essential steps:

• Regular testing: Use test kits to monitor chlorine levels, as they can fluctuate due to weather, usage, and dissipation. Aim for a consistent range to ensure swimmer comfort and safety.
• Scheduled chlorination: Based on chlorine testing results, add the needed amount of chlorine to offset the natural dissipation over time.
• Monitoring environmental factors: Sunlight and heat can increase the dissipation rate; thus, pools may need more frequent chlorination during hot, sunny periods.

Implementing a routine ensures chlorine stays within the desired range, between 1 ppm to 3 ppm, which is crucial for maintaining water clarity and hygiene.

Parts Per Million

Parts per million (ppm) is a measurement unit used to express very dilute concentrations of substances. In pool maintenance, it's crucial for measuring chlorine concentration, where it dictates safe and effective levels.
1 ppm indicates one part of chlorine per million parts of water. For pool water sanitation, maintaining a balance between 1 ppm and 3 ppm ensures that the water is both effective at disinfecting and comfortable for swimmers, as higher concentrations might cause irritation.
Understanding ppm for a pool involves:

• Using a precise test kit designed to measure ppm levels accurately.
• Recognizing that environmental changes and pool usage can affect these readings very quickly.
• Adjusting the concentration based on accurate ppm readings to align with safety standards.

Accurate ppm measurements and adjustments are the cornerstone of effective pool management.

Algae Growth Prevention

To prevent unwanted algae growth, maintaining proper chlorine levels is vital, which correlates directly with pool health and safety. Algae can turn water green, making it unsightly and potentially unsafe.
Algae thrives in environments where chlorine levels fall below 1 ppm, as stated in our exercise. This is because without adequate chlorine, microorganisms like algae can rapidly multiply, leading to discoloration and contamination of the pool water.
To combat algae growth, consider the following:

• Chlorine consistency: Keep chlorine at a minimum of 1 ppm, but ideally closer to 3 ppm for effective prevention.
• Physical maintenance: Regular cleaning and scrubbing of pool surfaces to prevent algae adherence.
• Prompt treatment: At the first sign of algae, apply an algicide or shock treatment combined with correct chlorination to swiftly rectify the balance.

Preventing algae is mainly about maintaining the right chemical environment in your pool, assisted by regular checks and physical maintenance.