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The problem asks to find the center of mass of a region of constant density bounded by the paraboloid \( z = 49 - x^2 - y^2 \) and the \( xy \)-plane.

### Steps to Solve

1. **Equation of the Paraboloid:**
   The equation \( z = 49 - x^2 - y^2 \) represents an upward-facing paraboloid. The region is bounded by this surface and the \( xy \)-plane (where \( z = 0 \)).

2. **Limits of Integration:**
   The paraboloid touches the \( xy \)-plane when \( z = 0 \), which gives:
   \[
   0 = 49 - x^2 - y^2 \quad \Rightarrow \quad x^2 + y^2 = 49.
   \]
   This is a circle of radius 7 centered at the origin in the \( xy \)-plane.

3. **Center of Mass Formula:**
   The center of mass coordinates for a solid of constant density can be calculated as:
   \[
   \bar{x} = \frac{1}{M} \int\int\int x \, dV, \quad \bar{y} = \frac{1}{M} \int\int\int y \, dV, \quad \bar{z} = \frac{1}{M} \int\int\int z \, dV,
   \]
   where \( M \) is the total mass of the region.

4. **Symmetry:**
   The region is symmetric about the \( z \)-axis, and since the density is constant, we conclude:
   \[
   \bar{x} = 0, \quad \bar{y} = 0.
   \]
   So, the problem reduces to finding \( \bar{z} \), the height of the center of mass along the \( z \)-axis.

5. **Volume Element in Cylindrical Coordinates:**
   To make the computation easier, switch to cylindrical coordinates:
   \[
   x = r \cos\theta, \quad y = r \sin\theta, \quad z = z, \quad dV = r \, dz \, dr \, d\theta.
   \]
   The equation of the paraboloid becomes:
   \[
   z = 49 - r^2.
   \]

6. **Mass and \( \bar{z} \):**
   To find \( \bar{z} \), compute the total mass \( M \) and the moment in the \( z \)-direction using the integrals over the volume.

### Answer:
The correct \( \bar{z} \) is found to be \( \frac{49}{3} \), making the center of mass coordinates \( \left( 0, 0, \frac{49}{3} \right) \).

Thus, the correct choice is:

\[
\boxed{\bar{x} = 0, \, \bar{y} = 0, \, \bar{z} = \frac{49}{3}}.
\]

Would you like further explanation on how the integrals are computed?

### Follow-up Questions:
1. How do you set up the mass integral for the region?
2. Can the problem be solved in rectangular coordinates instead?
3. What is the significance of symmetry in simplifying center of mass calculations?
4. How is the volume of the paraboloid computed?
5. What role does the density function play if it were not constant?

### Tip:
Always check for symmetry in center of mass problems, as it can greatly simplify calculations.

Find the center of mass of the region of constant density bounded by the paraboloid z = 49 - x^2 - y^2 and the xy-plane.

Learn how to find the center of mass of a region bounded by the paraboloid z = 49 - x^2 - y^2 and the xy-plane, using concepts of cylindrical coordinates and symmetry in calculus.

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