Question

1. the surface $\

ho = 4\cos \phi$ describes a sphere. identify the center and radius of the sphere. rewrite the equation of the sphere in cartesian (rectangular) and cylindrical coordinates.

Explanation:

Step1: Convert to spherical to Cartesian

Recall spherical to Cartesian conversions: $
ho^2 = x^2 + y^2 + z^2$, $z =
ho\cos\phi$. Multiply both sides of $
ho = 4\cos\phi$ by $
ho$:

$$ ho^2 = 4 ho\cos\phi$$

Substitute the conversions:
$$x^2 + y^2 + z^2 = 4z$$

Step2: Complete the square for z

Rearrange and complete the square for the z-terms:
$$x^2 + y^2 + z^2 - 4z = 0$$
$$x^2 + y^2 + (z^2 - 4z + 4) = 4$$
$$x^2 + y^2 + (z - 2)^2 = 2^2$$

Step3: Identify center and radius

From the standard sphere equation $(x-a)^2+(y-b)^2+(z-c)^2=r^2$, the center is $(a,b,c)$ and radius $r$.

Step4: Convert to cylindrical coordinates

Recall cylindrical to Cartesian conversions: $r^2 = x^2 + y^2$, $z=z$. Substitute into the Cartesian sphere equation:
$$r^2 + (z - 2)^2 = 4$$
Or expand from the intermediate step:
$$r^2 + z^2 = 4z$$

Answer:

• Center of the sphere: $(0, 0, 2)$
• Radius of the sphere: $2$
• Cartesian coordinate equation: $x^2 + y^2 + (z - 2)^2 = 4$
• Cylindrical coordinate equation: $r^2 + z^2 = 4z$ (or $r^2 + (z - 2)^2 = 4$)