MATH 12 Week3 ratio

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  • Week 3 Day 1 Ratio, Variation and Proportion (Algebra and Trigonometry, Young 2nd Edition, page 304-313)
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  • MATH 12 Week3 ratio

    1. 1. RATIO, VARIATION AND PROPORTION <br />MATH10 <br />ALGEBRA<br />Week 3 Day 1 Ratio, Variation and Proportion (Algebra and Trigonometry, Young 2nd Edition, page 304-313) <br />
    2. 2. Week 3 Day 1<br />TODAY’S OBJECTIVE<br />At the end of the lesson the students are expected to:<br />Use ratio and proportion in solving problems involving them,<br />Identify the different types of variation,<br />Understand the difference between direct variation and inverse variation,<br />Understand the difference between combined variation and joint variation, and <br />Develop mathematical models using direct variation, inverse variation, combined variation and joint variation.<br />
    3. 3. Week 3 Day 1<br />Definition <br />RATIO<br />A ratio is an indicated quotient of two quantities. Every ratio is a fraction and all ratios can be described by means of a fraction. The ratio of x and y is written as x : y, it can also be represented as <br /> Thus, <br />
    4. 4. Week 3 Day 1<br />EXAMPLE<br />1. Express the following ratios as simplified fractions:<br /> a) 5 : 20 <br /> b) <br />2. Write the following comparisons as ratios reduced to lowestterms. Use common units whenever possible.<br />a) 4 students to 8 students <br />b) 4 days to 3 weeks <br />c) 5 feet to 2 yards <br />d) About 10 out of 40 students took Math Plus <br />
    5. 5. Week 3 Day 1<br />Definition <br />PROPORTION<br />A proportion is a statement indicating the equality of two ratios. <br />Thus, , , are proportions.<br />In the proportion x : y = m : n, x and n are called the extremes, y and m are called the means.x and m are the called the antecedents, y and n are called the consequents.<br />In the event that the means are equal, they are called the mean proportional.<br />
    6. 6. Week 3 Day 1<br />EXAMPLE<br />1. Find the mean proportional of <br />2. Determine the value of x in the following proportion:<br /> a) 2 : 5 = x : 20 <br /> b) <br />
    7. 7. Week 3 Day 1<br />Definition<br />VARIATION<br />A variation is the name given to the study of the effects of changes among related quantities.<br />Variation describes the relationship between variables.<br />
    8. 8. Week 3 Day 1<br />Direct Variation<br />When one quantity is a constant multiple of another quantity, we say that the quantities are directlyproportional to one another .<br />Let x and y represent two quantities. The following are equivalent statements:<br /><ul><li> y = kx, where k is a nonzero constant.
    9. 9. y varies directly with x.
    10. 10. y is directly proportional to x.</li></ul>The constant k is called the constant of variation or the constant of proportionality.<br />Definition page 304<br />
    11. 11. Week 3 Day 1<br />EXAMPLE<br />Write an equation that describes each variation.<br />d is directly proportional to t. d=r when t=1.<br />V is directly proportional to both l and w.V=6h when w=3 qndh=4.<br />24. W is directly proportional to both R and the square of I. W=4 when R=100 and I=0.25.<br />(Exercises page 309)<br />
    12. 12. Week 3 Day 1<br />EXAMPLE<br />In the United States, the costs of electricity is directly proportional to the number of kilowatt hours (kWh) used. If a household in Tennessee on average used 3098 kWh per month and had an average monthly electric bill of $179.99, find a mathematical model that gives the cost of electricity in Tennessee in terms of the number of kWh used.(Example 1 page 304)<br />2. Hooke’s Law states that the force needed to keep a spring stretched x units beyond its natural length is directly proportional x. Here the constant of proportionality is called a spring constant.<br /> Write Hooke’s Law as an equation.<br /> If a spring has a natural length of 10 cm and a force of 40 N is required to maintain the spring stretched to a length of 15 cm, find the spring constant.<br />What force is needed to keep the spring stretched to a length of 14cm? ( Exercise 23 page 191 from Algebra & Trig. by Stewart, Redlin & Watson, 2nd edition)<br />
    13. 13. Week 3 Day 1<br />Direct Variation with Powers<br />Let x and y represent two quantities. The following are equivalent statements:<br /><ul><li>, where k is a nonzero constant.
    14. 14. y varies directly with the nth power of x.
    15. 15. y is directly proportional to the nth power of x.</li></ul>Definition page 305<br />
    16. 16. Week 3 Day 1<br />EXAMPLE<br />A brother and sister have weight (pounds) that varies as the cube of the cube of height (feet) and they share the same proportionality constant . The sister is 6 feet tall and weighs 170 pounds. Her brother is 6’4” tall. How much does he weigh?<br />(Your Turn page 306)<br />
    17. 17. Week 3 Day 1<br />Inverse Variation<br />Let x and y represent two quantities. The following are equivalent statements:<br /><ul><li> , where k is a nonzero constant.
    18. 18. yvaries inversely with x.
    19. 19. y is inversely proportional to x.</li></ul>The constant k is called the constant of variation or the constant of proportionality.<br />Definition page 306<br />
    20. 20. Week 3 Day 1<br />EXAMPLE<br />The number of potential buyers of a house decreases as the price of the house increases (see the graph on the below). If the number of potential buyers of a house in a particular city is inversely proportional to the price of the house, find a mathematical equation that describes the demand for the houses as it relates to the price. How many potential buyers will there be for a $2 million house? (Example 3 page 306)<br />(100,1000)<br />1000<br />800<br />Demand (number of potential buyers)<br />600<br />(200,500)<br />400<br />(400,250)<br />200<br />(600,167)<br />200<br />600<br />800<br />400<br />Price of the house (in thousands of dollars)<br />
    21. 21. Week 3 Day 1<br />Inverse Variation with Powers<br />Definition page 307<br />
    22. 22. Week 3 Day 1<br />Joint Variation and Combined Variation<br /><ul><li>When one quantity is proportional to the product of two or more other quantities, the variation is called joint variation.</li></ul>Example: Simple interest which is defined as<br /><ul><li>When direct variation and inverse variation occur at the same time, the variation is called combined variation.</li></ul>Example: Combined gas law in chemistry, <br />Definition page 307<br />
    23. 23. Week 3 Day 1<br />EXAMPLE<br />The gas in the headspace of a soda bottle has a volume of 9.0 ml, pressure of 2 atm (atmospheres), and a temperature of 298K (standard room temperature of 77⁰F). If the soda bottle is stored in a refrigerator, the temperature drops to approximately 279K (42⁰F). What is the pressure of the gas in the headspace once the bottle is chilled?<br />(Example 4 page 308)<br />
    24. 24. Week 3 Day 1<br />SUMMARY<br /> Direct, inverse, joint and combined variation can be used to model the relationship between two quantities. For two quantities x and y we say that:<br />Joint variation occurs when one quantity is directly proportional to two or more quantities.<br />Combined variation occurs when one quantity is directly proportional to one or more quantities and inversely proportional to one or more other quantities.<br />
    25. 25. Week 3 Day 1<br />CLASSWORK <br />#s page 20, 27,46,53 page 309-310<br />HOMEWORK <br />#s 22, 32, 33,36, 37, 39,40,42,43,47 page 309-313<br />
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