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Variation, Direct and Inverse Variation, Direct and Inverse

A variable is something that varies among components of a set or population, such as the height of high school students. Two types of relationships between variables are direct and inverse variation. In general, direct variation suggests that two variables change in the same direction. As one variable increases, the other also increases, and as one decreases, the other also decreases. In contrast, inverse variation suggests that variables change in opposite directions. As one increases, the other decreases and vice versa. Consider the case of someone who is paid an hourly wage. The amount of pay varies with the number of hours worked. If a person makes $12 per hour and works two hours, the pay is $24; for three hours worked, the pay is $36, and so on. If the number of hours worked doubles, say from 5 to 10, the pay doubles, in this case from $60 to $120. Also note that if the person works 0 hours, the pay is $0. This is an important component of direct variation: When one variable is 0, the other must be 0 as well. So, if two variables vary directly and one variable is multiplied by a constant, then the other variable is also multiplied by the same constant. If one variable doubles, the other doubles; if one triples, the other triples; if one is cut in half, so is the other. Algebraically, the relationship between two variables that vary directly can be expressed as y = kx, where the variables are x and y, and k represents what is called the constant of proportionality. (Note that this relationship can also be expressed as or = with also representing a constant.) In the preceding example, the equation is y 12x, with x representing the number of hours worked, y representing the pay, and 12 representing the hourly rate, the constant of proportionality. Graphically, the relationship between two variables that vary directly is represented by a ray that begins at the point (0, 0) and extends into the first quadrant. In other words, the relationship is linear, considering only positive values. See part (a) of the figure on the next page. The slope of the ray depends on the value of k, the constant of proportionality. The bigger k is, the steeper the graph, and vice versa. When two variables vary inversely, one increases as the other decreases. As one variable is multiplied by a given factor, the other variable is divided by that factor, which is, of course, equivalent to being multiplied by the reciprocal (the multiplicative inverse) of the factor. For example, if one variable doubles, the other is divided by two (multiplied by one-half); if one triples, the other is divided by three (multiplied by one-third); if one is multiplied by two-thirds, the other is divided by two-thirds (multiplied by three-halves). Consider a situation in which 100 miles are traveled. If traveling at an average rate of 5 miles per hour (mph), the trip takes 20 hours. If the average rate is doubled to 10 mph, then the trip time is halved to 10 hours. If the rate is doubled again, to 20 mph, the trip time is again halved, this time to 5 hours. If the average rate of speed is 60 mph, this is triple 20 mph. Therefore, if it takes 5 hours at 20 mph, 5 is divided by 3 to find the travel time at 60 mph. The travel time at 60 mph equals , or 1⅔ hours. Algebraically, if x represents the rate (in miles per hour) and y represents the time it takes (in hours), this relationship can be expressed as xy = 1000 or or . In general, variables that vary inversely can be expressed in the following forms: , or . The graph of the relationship between quantities that vary inversely is one branch of a hyperbola . See part (b) of the figure. The graph is asymptotic to both the positive x -and y -axes. In other words, if one of the quantities is 0, the other quantity must be infinite. For the example given here, if the average rate is 0 mph, it would take forever to go 100 miles; similarly, if the travel time is 0, the average rate must be infinite. Many pairs of variables vary either directly or inversely. If variables vary directly, their quotient is constant; if variables vary inversely, their product is constant. In direct variation, as one variable increases, so too does the other; in inverse variation, as one variable increases, the other decreases. see also Inverses; Ratio, Rate, and Proportion. Bob Horton Foster, Alan G., et al. Merrill Algebra 1: Applications and Connections. New York: Glencoe/McGraw Hill, 1995. ———. Merrill Algebra 2 with Trigonometry Applications and Connections. New York: Glencoe/McGraw Hill, 1995. Larson, Roland E., Timothy D. Kanold, and Lee Stiff. Algebra I: An Integrated Approach. Evanston, IL: D.C. Heath and Company, 1998.

From Omilili

Programming exercises

You will be writing a program that solves equations. The equation you will be using is d = vi * t that might have questions. The four basic kinematic equations are: Kinematic Equations and Problem-Solving... numbers, and for quadraticequations should highlight the line of symmetry and the vertex

From Yahoo Answers

Question:This is my second time posting this question. I have yet to find anyone who can lend any kind of real insight. Can you help? Some have argued, simplistically, that the core response for a direct mail publication should be the number of weeks it takes to achieve 80% of TOTAL response (following the simplistic 80/20 heuristic). Others have suggested that the core response rate should be time it takes to get to 90% to 100% of the total response (which could take infinitely long). Frequently, when looking at response data, it can take as many weeks to get from 80% to 90% as it did to get from 0% to 80%. This velocity of response varies with each direct mailing My thought is that velocity should be an important factor in this analysis, but I m ignorant about how to use it. Even 80% may not be appropriate depending upon how slow the response is. What is the best way to quantitatively define what a core response rate is, regardless of the velocity of response?

Answers:Went out on some Internet research. Unless you have a text-book definition there doesn't seem to be any consistent understanding about what it is. Available definitions seem purpose driven. Maybe if we knew how you wanted to use the "core response" somebody could figure out what a meaningful definition might be. Otherwise, it might be a "head" thing. Experience with the type of response (e.g. past statistics) could give an experienced person insight into the current - ongoing direct mail campaign. Alternatively, if the past results are available, a study of 'critical points' and the 'correctness' of decisions made at those points might give a template of key issues. However, I suspect Direct mail responses are finicky, subject to the winds of the economy and current fads. I don't think it's basically a math question. Maybe Business and Finance?

Question:I've been hearing over and over that what matters in the end is the amount of air displacement you have for how loud it will get. Like, how two 12s would be louder than one 15 because of the larger cone area displacing more air. So would a subwoofer with 0.1 cu. ft. air displacement be twice as loud as one woofer that has 0.05 cu. ft. of displacement? Is this logic correct? (Yes, I am aware that how loud a system is depends on a number of factors, but I am focusing on this one in particular.)

Answers:The air "displacement" you're referring to has to do with the physical size (and shape) of a specific driver and does not affect the SPL of the driver. A subwoofer with 0.1 Ft^3 displacement would simply be more massive than one displacing 0.05 Ft^3. The number is most useful in determining the final size of enclosure in which to mount the subwoofer. The displacement figure is subtracted from the enclosure's "gross" volume (along with anything else that takes up airspace - like a port for example) to arrive at the "net" volume. The volume of air "movement" produced by the subwoofer has a greater effect on the SPL. And air movement is more affected by the total cone area and the distance in and out that the cone is designed to move. "Xmax" specifies the distance the voice coil (and consequently the speaker cone) can move in one direction and generally the greater the Xmax, the more air is moved. But probably more important than either displacement or Xmax, and perhaps the best indication of SPL is the sensitivity, also known as "SPL". A higher sensitivity means the subwoofer is more efficient at turning electrical energy into useful audio. This figure shows the loudness in dB measured with 1 watt of power from a distance 1 meter directly in front of the speaker cone. Each time you double the power, the dB's increase by 3. So if you are comparing competing subwoofers and want to know which is "louder", with the other parameters constant, it's the one with the highest sensitivity.

Question:Sounds waves are longitudinal. Which DIRECTION are the VIBRATIONS in a longitudinal wave, compared to the direction the wave is traveling? Also, waht is meant by the frequency of a longitudinal/sound wave? thank you in advance.

Answers:Think it through: Sound is compression/rarefaction traveling along the axis of motion, right? The frequency will be the number of full cycles passing a given point/sec.

From Youtube

Burning Calories : Calculating Calories Burned in Exercise

The number of calories burned during exercise depends on a person's weight and the type of exercise. Learn more about calculating the calories burned in exercises with tips from a fitness trainer in this free weight loss video. Expert: Mike Quebec Bio: Mike Quebec is a certified trainer from the North American Sport Savate Association of American boxing and kickboxing. Filmmaker: Bing Hu


"Jailhouse" Tai Chi Form, red book on head, blindfold, front view - 08/26/09 (Summer) In this video, I am balancing a red book on my head throughout the entire "Jailhouse" Tai Chi Form, my compact adaptation of Cheng Manching's 37 Posture Tai Chi Form I'm also wearing a blindfold throughout the Form. ** A book balanced on my head is much more likely to stay in place throughout the entire Tai Chi Form if I initiate all turning movements very close to the central axis of my body. If I turn from my center, the book balanced on my head will turn together with my body, and will not fly off in one direction or another. It will just sit on my head and "go along for the ride." However, if I change directions suddenly, abruptly, in a jerky manner, then the book is much more likely to slide off my head, because it's being whipped around by unbalanced, clumsy forces vibrating up my spinal column. The smoother the turning action, the less likely a jerky motion will vibrate up my spine and shake the book off my head. An upright spine acts like a coiled spring / shock absorber. All the vertebrae and discs between them act like a vibrating medium. The head sits on top of this spring-like structure. If too much vibrational energy is allowed to pass up the spine, it will rock the head (like a whiplash), and shake the delicately balanced book off my head. By dropping my mind into my abdominal tan-tien, and moving smoothly from my pelvic center, I can ...

Strength Stretch Core Exercise Video Workout is a core workout fitness video focusing on your abdominal and back muscles, downloadable from this site directly to your hard drive. Preceded by a comprehensive series of stretches covering all major muscle groups, the routine utilizes the best parts of Yoga, Pilates, and Isometrics. The 50 minute routine is easy to follow for beginners, while at the same time providing a tough core workout for the experienced.

Upper Body Medicine Ball Exercises Power Drops

This medicine ball exercise is a variation of the chest pass movement. In this case the ball is dropped from a height to allow the athlete to yield to the mass of the ball and change direction to push the ball back to the partner. This makes this activity not only a medicine ball training exercise but also an upper body plyometric drill. For more medicine ball exercises, and training ideas visit

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