6.4 Variation 321 When you set up your new surround sound stereo system, you place a microphone at the place you typically sit when watching television. The computer within the receiver then generates a series of sounds, and automatically adjusts the volume of all the speakers in the system to give the best balance of sounds from the speakers. Because the sound you hear from each speaker is dependent on (or varies with) your distance from the speaker, the computer is using a concept called variation when activating your sound system. Variation SECTION 6.4 LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand direct variation. 7 Understand inverse variation. 7 Understand joint variation. 7 Understand combined variation. Why This Is Important Throughout this section we will use the concept of variation to help us model a wide variety of real-life applications. These applications involve medicine, geometry, economics, physics, chemistry, and music. Having knowledge of variation is a powerful problem-solving tool that we can use in many aspects of our lives. Direct Variation Many scientific formulas are expressed as variation equations. A variation equation is an equation that relates one variable to one or more other variables through the operations of multiplication or division (or both operations). There are essentially four types of variation problems: direct, inverse, joint, and combined variation. With direct variation, the values of the two related variables increase together or decrease together; that is, as one increases so does the other, and as one decreases so does the other. Consider a car traveling at 40 miles an hour. The car travels 40 miles in 1 hour, 80 miles in 2 hours, and 120 miles in 3 hours. Note that, as the time increases, the distance traveled increases, and, as the time decreases, the distance traveled decreases. Bao and Fong are married. Bao’s taxable income for the year was $34,200 and Fong’s was $36,400. Their total tax deduction for the year was $3640. This deduction can be divided between Bao and Fong any way they wish. How should the $3640 be divided between them to result in each individual having the same taxable income and therefore the greatest tax refund? Deduct $720 from Bao’s income and $2920 from Fong’s income. 53. Auto Insurance A driver education course at the East Lake School of Driving costs $45 but saves those under 25 years of age 10% of their annual insurance premiums until they are 25. Dan has just turned 18, and his insurance costs $100.00 per month. a) When will the amount saved from insurance equal the price of the course? 4.5 months b) Including the cost of the course, when Dan turns 25, how much will he have saved? $795 $ 54. Thermometer Comparison The relationship between Fahrenheit temperature (F) and Celsius temperature (C) is shown by the formula F C 9 5 32. = + At what temperature will a Fahrenheit thermometer read the same as a Celsius thermometer? 40° − Research Activities 55. Ratio and Proportion Ratio and proportion are used in many different ways in everyday life. Submit two articles in which ratios and/or proportions are used. Write a brief summary of each article explaining how ratio and/or proportion were used. 56. Nielsen Ratings Nielsen Media Research estimates the number of people who watch a television show—either from live television or from streaming services. Do research on the methods used by the Nielsen Ratings and write a report on your findings. Include how ratio and proportion are used to make their estimates. Andrey_Popov/ Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=