24–26. Sets in polar coordinates Sketch the following sets of ...

Question

24–26. Sets in polar coordinates Sketch the following sets of points.

4 ≤ r² ≤ 9

Accepted Answer

Welcome back, everyone. Sketch the set of points X Y in the plane that satisfy the inequality 16 is less than or equal to R2, which is less than or equal to 25. So for this problem, let's remember that in polar coordinates are squared is equal to x2 + Y2. So this inequality and Cartesian coordinates can be written as. X2 + Y2 is between 16 and 25 inclusive. What we're going to do is simply break this compound inequality down into two separate inequalities. So the first one is X2 + Y2 is greater than or equal to 16. What we can do is first we'll plot a circle in the form of x2 + y2 equals 16, right? Now remember on the right hand side of this equation we have the square of the radius, so R2d is equal to 16, which means that the radius of the circle is going to be 4. Now, this circle is centered at the origin, right, and we essentially want to plot a circle of radius form. So we're going to label. 4 points on the x axis we can choose 4 and -4. On the y axis once again we're choosing 4 and -4. Now we can connect these points to plot a circle of radius. Or centered at the origin, we can also increase precision slightly. And from here. We want to focus on the 2nd. Inequality, right? For the second inequality, the main difference is. Within the sign. So let's label it as number 2. In particular, we have X2 + Y2. It is less than or equal to 25, right? So far, we're not considering less than. Or equal to or greater than or equal to to plot. All of the solutions, right? All that we have to do is simply focus on the two circles themselves. So for this second part, let's go ahead and plot Y2 + X2 equals 25. So in this case, we have a circle of radius of square root of 25 is 5, right? So now what we want to do is simply Plot a circle centered at the origin. That has a radius of 5. We're going to label 5 and -5 on the Y axis, 5 and -5 on the X-axis, and we're going to connect these four points. Let's go ahead and do that. We now have a rough sketch of that circle, and now we can label all of the points that satisfy the actual compound inequality. What we can do is simply understand. That the inequalities signs will tell us precisely what region we want to shade. So essentially, based on the first inequality, it says that X2 + Y2 is greater than or equal to 16. So essentially any point above or below the circle, meaning outside of that circle that extends past the radius, right, satisfies the inequality. And for the second inequality, X2 + 52 must be less than or equal to 25. So it basically means that our solutions must be within the second circle, which has a larger radius, meaning all of the possible solutions are basically bounded by these two concentric circles, right? So we're going to label all of the region between these two circles. And we also want to ensure that each circle is drawn using a solid line, right? Because We have inclusive solutions, meaning less than or equal to or greater than or equal to. So there are subtleties such as ensuring that our circles are drawn using solid lines, right? And all of the solutions are basically between the two circles. That would be our final answer and thank you for watching.

24–26. Sets in polar coordinates Sketch the following sets of points.

4 ≤ r² ≤ 9

Key Concepts

Polar Coordinates System

Inequalities Involving r²

Graphing Regions in Polar Coordinates

Watch next