Test

by

Water Purification and Water Supply

1. Importance of Clean Water

  1. Essential for Health: Clean water is necessary for human well-being and daily activities.
  2. Purification Purpose: Removes impurities, microorganisms, odors, tastes, and colors to make water safe.
  3. Natural Contaminants: Water from natural sources often contains harmful substances that require treatment.

2. Methods of Water Purification

Filtration

  1. Removes Large Particles: Eliminates debris and suspended solids.
  2. Limitations: Cannot remove microorganisms, dissolved substances, or fine particles.
  3. Sand Filtration: Helps remove smaller suspended materials.

2.2 Boiling

  1. Kills Microorganisms: Heating water to 100°C eliminates bacteria and viruses.
  2. Limitations: Does not remove dissolved substances or solid impurities.

2.3 Chlorination

  1. Kills Microorganisms: Chlorine is added to disinfect water.
  2. Application: Commonly used in liquid form.
  3. Limitations: Does not remove dissolved or suspended substances; excessive chlorine may cause an unpleasant odor and health risks.

2.4 Distillation

  1. Produces Pure Water: Removes all suspended and dissolved substances along with microorganisms.
  2. Process: Boiling water and condensing the steam for collection.
  3. Limitations: Lacks essential minerals, making it less ideal for drinking.

3. Water Supply System

  1. Water Sources: Rivers, streams, and reservoirs provide raw water.
  2. Treatment Process: Water is purified before distribution.
  3. Distribution Network: Clean water is supplied through pipelines.

3.1 Main Components

  1. Dams: Store and regulate water resources.
  2. Treatment Plants: Purify water before distribution.
  3. Pipe Network: Delivers clean water to consumers.

4. Stages in Water Treatment

4.1 Screening/Filtration

  1. Removes Debris: Eliminates large particles like leaves and branches.

4.2 Oxidation

  1. Enhances Quality: Introduces oxygen to eliminate odors and improve taste.

4.3 Coagulation

  1. Particle Removal: Alum clumps suspended particles for easier removal.
  2. Acidity Control: Lime neutralizes excess acidity caused by alum.

4.4 Sedimentation

  1. Settling Process: Clumped particles settle at the bottom and are removed.

4.5 Sand Filtration

  1. Fine Particle Removal: Water passes through sand layers to filter out remaining impurities.

4.6 Chlorination

  1. Disinfection: Chlorine is added to kill any remaining microorganisms.

4.7 Fluoridation

  1. Dental Health: Fluoride is introduced to prevent tooth decay.

4.8 Storage

  1. Final Step: Treated water is stored in tanks before distribution.

5. Alternative Water Sources

5.1 Water Recycling

  1. Reuses Wastewater: Treats wastewater for domestic, industrial, and drinking purposes.
  2. Treatment Stages: Primary (physical), secondary (biological), and tertiary (chemical).

5.2 Fog Collection

  1. Captures Water from Fog: Uses polypropylene nets to collect water droplets for use.

5.3 Desalination

  1. Removes Salt from Seawater: Uses reverse osmosis and semi-permeable membranes to produce freshwater.

6. Water Sustainability

6.1 Water Pollution

  1. Causes: Domestic waste, agriculture, industrial discharge, and urban development.
  2. Pollutants: Includes silt, pesticides, fertilizers, oil spills, radioactive waste, and heavy metals like mercury.
  3. Effects: Leads to algal blooms, marine life damage, disease outbreaks, and contamination risks.

6.2 Protecting Water Sources

  1. Public Education: Raise awareness about pollution prevention.
  2. Improved Infrastructure: Upgrade sewage systems and sanitation facilities.
  3. Agricultural Practices: Promote biodegradable fertilizers and pesticides.
  4. Regulations: Enforce industrial waste treatment before disposal.

6.3 Individual Water Conservation

  1. Reduce Water Use: Take showers instead of baths and turn off taps while brushing.
  2. Efficient Washing: Use basins for dishwashing and buckets for car cleaning.
  3. Optimized Laundry: Run washing machines with full loads.
  4. Rainwater Collection: Store rainwater for gardening and other non-drinking uses.
  5. Fix Leaks Promptly: Prevent water wastage by repairing pipes.

**Detailed Explanation of Adding and Subtracting Surds** ### **Understanding the Concept** Surds are irrational numbers expressed in their root form, such as \( \sqrt{2} \) or \( \sqrt{5} \), which cannot be simplified into exact decimals. When adding or subtracting surds, specific rules must be followed to ensure correct simplification. ### **1. The Importance of Like Terms** Just as in algebra, you can only add or subtract surds if they are **like terms**—meaning they have the same number under the square root sign. – Example 1: \( 2\sqrt{5} + 7\sqrt{5} = 9\sqrt{5} \), since both terms contain \( \sqrt{5} \). – Example 2: \( 2\sqrt{5} + 7\sqrt{6} \) cannot be simplified further, as \( \sqrt{5} \) and \( \sqrt{6} \) are different surds. ### **2. The Need for Simplification** In many cases, surds must be simplified before they can be added or subtracted. This involves breaking them down into factors, identifying perfect square factors, and simplifying accordingly. #### **How to Simplify Surds Before Addition or Subtraction** 1. **Identify perfect square factors**: Find square numbers within the radicand (the number inside the square root). 2. **Rewrite the surd**: Express the surd in terms of the square root of its factors. 3. **Extract the square root of perfect squares**: Factor out and simplify. 4. **Combine like terms**: Add or subtract the surds if they are now similar. 5. **Leave unlike terms separate**: If surds remain different, they cannot be combined. ### **Examples of Adding and Subtracting Surds** #### **Example 1: Simplify \( 10\sqrt{2} – 7\sqrt{2} \)** Since both terms contain \( \sqrt{2} \), simply subtract the coefficients: \[ 10\sqrt{2} – 7\sqrt{2} = 3\sqrt{2} \] #### **Example 2: Simplify \( 6\sqrt{24} \)** \[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \] Now multiply by 6: \[ 6 \times 2\sqrt{6} = 12\sqrt{6} \] #### **Example 3: Simplify \( \sqrt{27} – \sqrt{12} \)** \[ \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \] \[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \] Subtracting: \[ 3\sqrt{3} – 2\sqrt{3} = \sqrt{3} \] #### **Example 4: Simplify \( \sqrt{63} + \sqrt{28} – \sqrt{175} \)** \[ \sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7} \] \[ \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} \] \[ \sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7} \] Now, add and subtract: \[ 3\sqrt{7} + 2\sqrt{7} – 5\sqrt{7} = 0 \] #### **Example 5: Simplify \( \sqrt{20} + \sqrt{45} + \sqrt{48} \)** \[ \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \] \[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \] \[ \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \] Combining like terms: \[ 2\sqrt{5} + 3\sqrt{5} + 4\sqrt{3} = 5\sqrt{5} + 4\sqrt{3} \] Since \( \sqrt{5} \) and \( \sqrt{3} \) are not like terms, they remain separate. ### **3. Rules to Remember When Working with Surds** – **Only like surds can be added or subtracted**. – **Always simplify surds first** if possible. – **Break down the radicand into perfect squares** to simplify further. – **Leave unlike surds as separate terms**. – **Follow basic surd rules**, such as \( a\sqrt{b} = \sqrt{a^2b} \), which helps in breaking down complex expressions. ### **Summary of Steps for Adding and Subtracting Surds** 1. **Check if the surds have the same radicand**. 2. **Simplify each surd individually** by breaking them into their prime factors. 3. **Extract perfect square factors** and simplify. 4. **Combine like surds by adding or subtracting coefficients**. 5. **Leave the final answer in its simplest form**. Mastering these techniques allows for efficient manipulation of surds in mathematical problems, including algebra, geometry, and higher-level mathematics.

Detailed Explanation of Adding and Subtracting Surds

Understanding the Concept

Surds are irrational numbers expressed in their root form, such as 2\sqrt{2} or 5\sqrt{5}, which cannot be simplified into exact decimals. When adding or subtracting surds, specific rules must be followed to ensure correct simplification.

1. The Importance of Like Terms

Just as in algebra, you can only add or subtract surds if they are like terms—meaning they have the same number under the square root sign.

  • Example 1: 25+75=952\sqrt{5} + 7\sqrt{5} = 9\sqrt{5}, since both terms contain 5\sqrt{5}.
  • Example 2: 25+762\sqrt{5} + 7\sqrt{6} cannot be simplified further, as 5\sqrt{5} and 6\sqrt{6} are different surds.

2. The Need for Simplification

In many cases, surds must be simplified before they can be added or subtracted. This involves breaking them down into factors, identifying perfect square factors, and simplifying accordingly.

How to Simplify Surds Before Addition or Subtraction

  1. Identify perfect square factors: Find square numbers within the radicand (the number inside the square root).
  2. Rewrite the surd: Express the surd in terms of the square root of its factors.
  3. Extract the square root of perfect squares: Factor out and simplify.
  4. Combine like terms: Add or subtract the surds if they are now similar.
  5. Leave unlike terms separate: If surds remain different, they cannot be combined.

Examples of Adding and Subtracting Surds

Example 1: Simplify 102−7210\sqrt{2} – 7\sqrt{2}

Since both terms contain 2\sqrt{2}, simply subtract the coefficients:

102−72=3210\sqrt{2} – 7\sqrt{2} = 3\sqrt{2}

Example 2: Simplify 6246\sqrt{24}

24=4×6=4×6=26\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}

Now multiply by 6:

6×26=1266 \times 2\sqrt{6} = 12\sqrt{6}

Example 3: Simplify 27−12\sqrt{27} – \sqrt{12}

27=9×3=33\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} 12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

Subtracting:

33−23=33\sqrt{3} – 2\sqrt{3} = \sqrt{3}

Example 4: Simplify 63+28−175\sqrt{63} + \sqrt{28} – \sqrt{175}

63=9×7=37\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7} 28=4×7=27\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} 175=25×7=57\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7}

Now, add and subtract:

37+27−57=03\sqrt{7} + 2\sqrt{7} – 5\sqrt{7} = 0

Example 5: Simplify 20+45+48\sqrt{20} + \sqrt{45} + \sqrt{48}

20=4×5=25\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} 45=9×5=35\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} 48=16×3=43\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}

Combining like terms:

25+35+43=55+432\sqrt{5} + 3\sqrt{5} + 4\sqrt{3} = 5\sqrt{5} + 4\sqrt{3}

Since 5\sqrt{5} and 3\sqrt{3} are not like terms, they remain separate.

3. Rules to Remember When Working with Surds

  • Only like surds can be added or subtracted.
  • Always simplify surds first if possible.
  • Break down the radicand into perfect squares to simplify further.
  • Leave unlike surds as separate terms.
  • Follow basic surd rules, such as ab=a2ba\sqrt{b} = \sqrt{a^2b}, which helps in breaking down complex expressions.

Summary of Steps for Adding and Subtracting Surds

  1. Check if the surds have the same radicand.
  2. Simplify each surd individually by breaking them into their prime factors.
  3. Extract perfect square factors and simplify.
  4. Combine like surds by adding or subtracting coefficients.
  5. Leave the final answer in its simplest form.

Mastering these techniques allows for efficient manipulation of surds in mathematical problems, including algebra, geometry, and higher-level mathematics.

Leave a Comment