We are asked to describe the graphs of the given cylindrical or spherical equations. Problem 7: $r = 5$ in cylindrical coordinates. Problem 8: $\rho = 5$ in spherical coordinates. Problem 9: $\phi = \pi/6$ in spherical coordinates. Problem 10: $\theta = \pi/6$ in cylindrical coordinates. Problem 11: $r = 3\cos\theta$ in cylindrical coordinates. Problem 12: $r = 2\sin 2\theta$ in cylindrical coordinates.

1. Problem Description

We are asked to describe the graphs of the given cylindrical or spherical equations.

Problem 7: $r = 5$ in cylindrical coordinates.

Problem 8: $\rho = 5$ in spherical coordinates.

Problem 9: $\phi = \pi/6$ in spherical coordinates.

Problem 10: $\theta = \pi/6$ in cylindrical coordinates.

Problem 11: $r = 3\cos\theta$ in cylindrical coordinates.

Problem 12: $r = 2\sin 2\theta$ in cylindrical coordinates.

2. Solution Steps

Problem 7: $r = 5$

In cylindrical coordinates, $r$ represents the distance from the z-axis. The equation $r = 5$ represents a cylinder of radius 5 centered on the z-axis.

Problem 8: $\rho = 5$

In spherical coordinates, $\rho$ represents the distance from the origin. The equation $\rho = 5$ represents a sphere of radius 5 centered at the origin.

Problem 9: $\phi = \pi/6$

In spherical coordinates, $\phi$ is the angle from the positive z-axis. The equation $\phi = \pi/6$ represents a cone opening upwards from the origin with a half-angle of $\pi/6$ with respect to the positive z-axis.

Problem 10: $\theta = \pi/6$

In cylindrical coordinates, $\theta$ is the angle from the positive x-axis in the xy-plane. The equation $\theta = \pi/6$ represents a half-plane that extends from the z-axis at an angle of $\pi/6$ from the positive x-axis.

Problem 11: $r = 3\cos\theta$

In cylindrical coordinates, we can multiply both sides by $r$ to get $r^2 = 3r\cos\theta$. We know that $r^2 = x^2 + y^2$ and $x = r\cos\theta$. So, the equation becomes $x^2 + y^2 = 3x$. Completing the square, we have $x^2 - 3x + (3/2)^2 + y^2 = (3/2)^2$, which simplifies to $(x - 3/2)^2 + y^2 = (3/2)^2$. This is a circle in the xy-plane centered at $(3/2, 0)$ with a radius of $3/2$. This extends infinitely along the z-axis forming a cylinder.

Problem 12: $r = 2\sin 2\theta$

In cylindrical coordinates, we can use the double angle formula to rewrite the equation as $r = 4\sin\theta\cos\theta$. Multiplying both sides by $r^2$, we get $r^3 = 4r\sin\theta r\cos\theta$. Since $x = r\cos\theta$, $y = r\sin\theta$, and $r^2 = x^2 + y^2$, we have $r = \sqrt{x^2 + y^2}$. Thus, $(x^2 + y^2)^{3/2} = 4xy$.

This represents a four-leaf rose in the xy-plane.

3. Final Answer

Problem 7: A cylinder of radius 5 centered on the z-axis.

Problem 8: A sphere of radius 5 centered at the origin.

Problem 9: A cone opening upwards from the origin with a half-angle of $\pi/6$ with respect to the positive z-axis.

Problem 10: A half-plane that extends from the z-axis at an angle of $\pi/6$ from the positive x-axis.

Problem 11: A cylinder whose cross-section in the xy-plane is a circle centered at (3/2, 0) with radius 3/

2. Problem 12: A four-leaf rose in the xy-plane, given by the equation $(x^2 + y^2)^{3/2} = 4xy$.