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Direct Variation - Concept
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
We often use the term direct variation to describe a form of dependence of one variable on another. An equation that makes a line and crosses the origin is a form of direct variation, where the magnitude of x increases or decreases directly as y increases or decreases. Direct variation and inverse variation are used often in science when modeling activity, such as speed or velocity.
A direct variation is when you have a relationship between two variables and they are on the same level. What I mean by that is there's no, their variables aren't divided by anything they're just equal to each other varying by a constant okay.
We have right here two examples that hopefully look familiar to you. Circumference of a circle is equal to 2 pi times the radius. What this is saying is the circumference is directly related to the radius just changing by a factor of 2 pi. When a radius gets bigger so is their circumference, when the circumference gets bigger so does the radius. There's a direct relationship between these two okay? They're off by a little number the circumference is always 2 pi bigger than the radius. This 2 pi is called the constant of variation okay? No matter what the circle no matter what the radius no matter what the circumference this 2 pie will always be there it's constant for this entire relationship okay.
Another example area of a circle is equal to pi r squared okay and this time the area is directly related to the radius squared, the radius gets bigger the area gets bigger, area gets bigger radius gets bigger there's a direct relationship between these two. This time our constant of variation is pi a little different than before but it's for every single area radius problem it's always going to be the same area is equal to pi r squared okay.
Let's look at another example of direct variation okay. This one is just into a hidden into a work problem. You work 10 hours and make a $150 okay. How much do you expect to make if you work for 12 hours. So just thinking about this logically, you work for 10 hours and make $150, how much do you make an hour? 150 you make divided by 10 means you make $15 an hour. You expect that to stay the same through out so you're always going to make $15 an hour this time you work 12 hours so you expect to make 12 hours of work times $15 which is just going to be $108. There's a direct relationship between the amount you work and the amount you get paid.
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Sample Problems (5)
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Problem 1 4,378 views
y varies directly as the cube of x.If y = 32 when x = 2, find y when x = 3.
Problem 2 3,503 views
A ball is dropped from the top of a building. The distance the ball falls is directly proportional to the square of the time it has been falling.If a ball falls 48 feet in 2 seconds, find how far it has fallen after 6 seconds.
Problem 3 131 views
Problem 4 106 views
Problem 5 119 views