How do you identify the surface represented by the given vector equation r(s, t) = (s * sin(9t), s^2, s * cos(9t))?

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The vector equation r(s, t) = (s * sin(9t), s^2, s * cos(9t)) defines a surface in three-dimensional space, where the parameters s and t vary.

To identify the surface, we can break down the components of the vector equation:

  1. x = s * sin(9t)
  2. y = s^2
  3. z = s * cos(9t)

### Step 1: Expressing in terms of s

The parameter s can be related to the y component, since we have y = s^2. This implies:

 s = ±√y

Substituting s back into the equations for x and z gives us:

 x = ±√y * sin(9t) 
 z = ±√y * cos(9t)

### Step 2: Finding a relationship between x, y, and z

Notice that both x and z can be expressed as:

 x^2 + z^2 = (±√y * sin(9t))^2 + (±√y * cos(9t))^2

Calculating this yields:

 x^2 + z^2 = y * (sin²(9t) + cos²(9t)) 
ightarrow x^2 + z^2 = y

### Conclusion

The final equation we derive is:

 x^2 + z^2 = y

This equation represents a vertical paraboloid surface in three-dimensional space, opening upwards along the y-axis. The parameter t allows for circular motion in the x-z plane, with frequency determined by the coefficient of t. Thus, the surface identified by the given vector equation is a parabolic shape revolving around the y-axis.

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