Carl Horowitz

**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!

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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.