Question

In Exercises \(35-40,\) find an equation in rectangular coordinates for the equation given in spherical coordinates, and sketch its graph. $$ \rho=4 \cos \phi $$

Short answer

The equation in rectangular coordinates is \(z = 4 - 4 \sqrt{x^2 + y^2}\). It represents a cone opening downwards with vertex at \(z = 4\), centered along the z-axis.

Step-by-step solution

Convert from Spherical to Rectangular Coordinates

We start first by using the relationship between rectangular and spherical coordinates: \(\rho=4 \cos \phi\), where \(z = \rho \cos{\phi}\). Therefore, we can say \(z = 4 \cos{\phi} \cos{\phi}\). This simplifies to \(z = 4 \cos^2{\phi}\). Since we know that \(\cos^2{\phi}\) can be replaced by \(1 - \sin^2{\phi}\), The equation becomes \(z= 4 (1 - \sin^2{\phi})\). Using the conversion relationship that \(\sin \phi = \sqrt{x^2 + y^2} / \rho\), we substitute this value in. Since \(\rho = 4 \cos \phi\) was given in the original equation, we can rewrite \(\rho\) in terms of \(z\), which is \(4 (1 - z^2)\). This makes our equation as \(z = 4 - 4 \sqrt{x^2 + y^2}\).

Sketch the Graph

To understand this graph, it is best to consider it in three dimensions. The equation \(z = 4 - 4 \sqrt{x^2 + y^2}\) describes a cone opening downwards in the negative z direction with a vertex at \(z = 4\). The cone is centered on the z axis and due to the form of the equation, we know that as we move away from the z-axis (increase \(x^2 + y^2\)), z decreases. We also note that the equation has reflectional symmetry about the z axis as the equation does not depend on the angle \(\theta\) around the z axis. A good sketch will therefore show this cone.