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2.10

UNSW Sydney

Skip to 0 minutes and 11 secondsSo now we're gonna have a look at quadratic relations, where one quantity is a fixed multiple of the square of another. So what is a quadratic relation? Well, here's the fundamental equation. y equals kx squared. We have two quantities, x and y, and they're related in this fashion. k is a fixed constant, like 2, or 3, or something like that. All right, so let's note that if we double x, then y is multiplied by a factor of 4. If we triple x, then y is multiplied by a factor of 9, and so on. That's independent of the constant k. So some important examples-- well, one is the relationship between area and side length. So here's a square, side length r.

Skip to 0 minutes and 59 secondsAnd the area of this square will be r squared. That's an example of a quadratic relation where the constant k is, of course, 1. What happens if we take an equilateral triangle with side lengths r? Well, we'd need to do a little bit of geometry to find the area. This is going to be r over 2. This height here is going to be root 3 over 2 times r. And so we can say that the area of the triangle, the equilateral triangle, will be 1/2 base times height. And so that will be root 3 over 4 times r squared. Another example of a quadratic proportionality, but now the constant is different than it is for the square.

Skip to 1 minute and 51 secondsAnd of course, an even more famous example is that of the circle. If we take the radius r there, then the area of a circle, is-- well, it's also a constant times r squared, but now the constant is this fabulous number, pi. Roughly 3.14, et cetera. So these are all examples of quadratic relations where the constants are notably different in all three cases. And now let's put these quadratic relations to work in understanding some important basic physics.

Quadratic relations here there and everywhere

Quadratic relations occur frequently in physics and economics. Here we get an introduction to the basic definition, and look at a simple but important family of examples from geometry.

The basic definition of a quadratic relation is a lot like that of a direct proportionality, except that one of the variables is squared. Thus $\normalsize{y=ax^2}$ is a typical quadratic relation. It has the property that if $\normalsize{x}$ is doubled, then $\normalsize{y}$ gets multiplied by four. If $\normalsize{x}$ is tripled, then $\normalsize{y}$ gets multiplied by nine. This kind of scaling is independent of the particular constant $\normalsize{a}$.

We are going to be looking at a range of examples, coming from physics and economics and also mathematics. The simplest examples relate area to linear measurements: area is intrinsically a quadratic quantity. If we look at the classical formulas for areas of squares, triangles and circles, we see interesting numbers appearing as constants of this (quadratic) proportionality.