The equation $$ z^{2}+y^{2}-( an (x))^{2}=0 $$ represents a surface of revolution. Around which axis was the profile curve revolved to generate this equation? $$ x $$-axis $$ y $$-axis Write the equation of the profile curve in the $$ x z $$-plane.

Question

The equation $$ z^{2}+y^{2}-(	an (x))^{2}=0 $$ represents a surface of revolution. Around which axis was the profile curve revolved to generate this equation? $$ x $$-axis $$ y $$-axis Write the equation of the profile curve in the $$ x z $$-plane.

UpStudy AI · Accepted Answer

The given equation is:

$$
z^{2} + y^{2} - (\tan(x))^{2} = 0
$$

This can be rewritten as:

$$
z^{2} + y^{2} = \tan^{2}(x)
$$

### Axis of Revolution

To determine around which axis the profile curve was revolved, let's analyze the equation:

- **Revolving Around the $$ x $$-axis:**
  
  If we have a profile curve in the $$ xz $$-plane described by $$ z = f(x) $$, revolving this curve around the $$ x $$-axis generates a surface where every point on the curve traces a circle in the $$ yz $$-plane. The equation of such a surface is:

$$
  y^{2} + z^{2} = f(x)^{2}
  $$

Comparing this with the given equation $$ y^{2} + z^{2} = \tan^{2}(x) $$, we see a direct match. This indicates that the profile curve $$ z = \tan(x) $$ in the $$ xz $$-plane was revolved around the $$ x $$-axis to generate the given surface.

- **Revolving Around the $$ y $$-axis:**
  
  Revolving a curve around the $$ y $$-axis would involve the $$ x $$ and $$ z $$ coordinates being expressed in terms of the radial distance from the $$ y $$-axis, which doesn't align with the given equation structure.

### Equation of the Profile Curve in the $$ xz $$-Plane

Given that the surface is generated by revolving around the $$ x $$-axis, the profile curve lies in the $$ xz $$-plane and is described by:

$$
z = \tan(x)
$$

### Conclusion

- **Axis of Revolution:** $$ x $$-axis
- **Profile Curve Equation in the $$ xz $$-plane:** $$ z = \tan(x) $$

**Final Answer:**

• The surface is obtained by rotating the profile curve about the x-axis.
• In the xz-plane, the profile curve is z = tan x.
The surface is generated by rotating the profile curve $$ z = \tan(x) $$ around the $$ x $$-axis.

The equation

represents a surface of revolution. Around which axis was the profile curve revolved to generate this
equation?
-axis
-axis
Write the equation of the profile curve in the -plane.

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The equation represents a surface of revolution. Around which axis was the profile curve revolved to generate this equation? -axis -axis Write the equation of the profile curve in the -plane.