In Exercises 19-24, match the parametric equations with the parametric curves labeled A through

In Exercises 19-24, match the parametric equations with the...

Key Concepts

Parametric Equations

Parametric equations describe a set of coordinates as functions of a common parameter, often used to express curves. This method allows for a clear representation of the movement and position on the curve, making it easier to analyze the behavior and shape of the curves as the parameter varies.

Conversion to Polar Coordinates

The given equations can be expressed in polar form by recognizing that x = r cos(?) and y = r sin(?). By comparing these with the equations x = (1/4)t cos(t) and y = (1/4)t sin(t), one can see that the parameter t plays the role of the angle while the radius r is proportional to t, facilitating the analysis of the curve's properties in the polar coordinate system.

Archimedean Spiral

An Archimedean spiral is characterized by a linear relationship between the radius and the angle, meaning the distance from the origin increases uniformly as the angle increases. In this context, the equations represent a spiral where r = (1/4)t, which is the hallmark of an Archimedean spiral, allowing for the identification of the curve by its ever-increasing radius as the parameter (or angle) increases.

Transcript

00:02 In this question, we are given a parametric equation, where x of t is a quarter t times close of t, and y of t equals a quarter t times sine of t.

00:24 So this is pretty close once again to the classical circle, except that the radius isn't a constant.

00:32 The radius of our circle is a quarter key.

00:37 So basically the radius of our circle is changing in time.

00:46 Now, this is going to be a spiral, and we're going to see that that is actually one of the options.

00:52 So essentially how you can think about this, we'll see the options in a bit, is the cosine sine part just rotates around like this.

01:06 Over the regular circle forever.

01:11 And this factor of a quarter t scales it.

01:14 So a t equal zero, we start at zero because the radius is zero.

01:19 And then we are rotating around, and at the same time, our radius is increasing.

01:27 So we are going to find some nice spiral like this.

01:32 So let's look at the options.

01:34 Here the options are in order, and we see that one of our options is in fact a spiral.

01:44 So this guy.

01:48 We can quickly go over why the other ones aren't it, which is also a good way to do this.

01:55 So looking at this blue one, this x function will at some point be negative, because the cosine will be negative.

02:09 So this is going to be negative and this is never negative.

02:15 This guy could be it, and again, the radius is evolving with time and not in a periodic manner...

Please give Ace some feedback