15–22. Sets in polar coordinates Sketch the following sets of ...

Question

15–22. Sets in polar coordinates Sketch the following sets of points.

2 ≤ r ≤ 8

Accepted Answer

Welcome back, everyone. Sketch the set of points on a Cartesian plane where the distance R from the origin satisfies the inequality of our being between 3 and 7 inclusive. For this problem, first of all, let's use the definition of R. and in particular, we know that R2d is equal to X2 + Y2. when we perform the conversion from polar coordinates to Cartesian coordinates. So, what we can do is simply manipulate the inequality because all sides are positive, we can just square all terms, right? So we get 3 squared is less than or equal to R2 and R2 is less than or equal to 72. We get 9 on the left hand side is less than or equal to R2, which can now be written as. X2 + y2. Which is less than or equal to 49. Now, what we're going to do is simply consider each side of this compound inequality, starting with the left hand side. Let's label it as number 1, and first of all, we're considering X2 + Y2 is greater than or equals to 9. Let's remember that. An equation in the form of X2 + y2 equals R2 represents a circle centered at diversion with a radius of R. So what we want to do for the first inequality is plot a circle in the form of x2 + y2 equals 9, and then we're going to label all of the points that satisfy this inequality, right? So they will be above the circle because we want to ensure that all of the points are. Basically satisfying the inequality greater than or equal to 9. To plot this inequality, or specifically the solutions to this inequality, we want to identify the radius and the radius would be 3 because we're taking square root of 9 and that's 3. And now what we can do is simply plot that circle. We know that the center is at the origin. We know that the radius is 3, so we're going to label. We can actually label 4 points, right? So one on the positive x-axis, that would be 3, then -3 on the negative X axis, 3 on the Y axis, and -3 on the negative Y axis. And now we can basically connect these four points in a circle. Well done. So now we just have that circle and we have to understand that the solutions to this first inequality. are above or below the circle. To state it more clearly, we are only considering the solutions that are outside of the circle, not inside, right? And we also want to ensure that the circle itself is plotted using a solid line because the inequalities are greater than or equal to, meaning we're also including the solutions that lie on the circle itself. Well done. And now, let's consider the second inequality because we want to identify the whole region. So that'll be #2. For number 2, we are considering X2 + Y2 is less than or equal to 49. And once again, we can use a circle in the form of X2 + Y2 equals 49. It would have a radius of square root of 49 which is 7. So we're going to plot a circle centered at the origin with a radius of 7. Let's add 4 points a game. Where X we take 77. When Y is 0, right? And when X is 0, we take Y of 7 and Y of -7. So, now that we have these 4 points, we're going to connect them. In a circle. Well done. Now let's ensure that the accuracy of our. Sketch Yes Appropriate, right? We want to ensure that it's crossing all of the correct points. OK. And now from here, we can essentially label the region that satisfies the, the original compound inequality. We want to ensure that The solutions to the compound inequality are. Between the two circles. Now why is that? Well, the first inequality says greater than or equal to 9, so we are outside of the smaller circle, but we are inside of the bigger circle. It says less than or equal to 49, so we can now shade all of the region between. These two circles. Let's go ahead and do that and let's ensure that each circle. Is drawn. In solid lines, right, because our inequality says greater than or equal to and less than or equal to. So basically all of the region between these two circles satisfies the inequality. We can essentially shade it. And All of that region. Is going to represent the solutions to our. Inequality So the whole idea in this problem was to. First we'll convert R. Into X2 + y2 raises the power of 1/2 or basically square root of X2 + y2. We used a fact that if we squarer, we get R2 equals X2 + y2, right? And then we used an equation of a circle centered at divergent to identify. The region of interest. Now that we have identified the region of interest, we can just shade the area. To clearly show. That all of our points of interest are between the two circles, right? We can essentially keep. Adding Green in this case to ensure that we can clearly see which region that is. Well, once we have an approximate. Re That would be enough to clearly show where the boundaries of that region. Are Let's label our final answer and thank you for watching.

15–22. Sets in polar coordinates Sketch the following sets of points.

2 ≤ r ≤ 8

Key Concepts

Polar Coordinates System

Inequalities in Polar Coordinates

Graphing Regions in Polar Coordinates

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