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To find the mass of the solid \( E \) below the plane \( z = 10 \) and above the cone \( z = \sqrt{x^2 + y^2} \) with density \( \rho(x,y,z) = z \), we need to set up and evaluate the triple integral in cylindrical coordinates.

### Conversion to Cylindrical Coordinates:
In cylindrical coordinates:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( z = z \)
- \( dV = r \, dr \, d\theta \, dz \)

### Solid \( E \):
- The cone \( z = \sqrt{x^2 + y^2} \) converts to \( z = r \).
- The plane \( z = 10 \) remains the same.
- Thus, \( r \leq z \leq 10 \).
- The radius \( r \) ranges from \( 0 \) to \( 10 \).
- The angle \( \theta \) ranges from \( 0 \) to \( 2\pi \).

### Triple Integral:
The mass of \( E \) is given by:
\[
\iiint_{E} \rho(x,y,z) \, dV = \iiint_{E} z \, dV
\]

In cylindrical coordinates, this becomes:
\[
\iiint_{E} z \cdot r \, dr \, d\theta \, dz
\]

### Integration Limits:
- \( 0 \leq \theta \leq 2\pi \)
- \( r \leq z \leq 10 \)
- \( 0 \leq r \leq 10 \)

### Setting up the Integral:
\[
\int_{0}^{2\pi} \int_{0}^{10} \int_{r}^{10} z \cdot r \, dz \, dr \, d\theta
\]

### Evaluate the Integral:
1. **Integrate with respect to \( z \)**:
\[
\int_{r}^{10} z \, dz = \left[ \frac{z^2}{2} \right]_{r}^{10} = \frac{10^2}{2} - \frac{r^2}{2} = 50 - \frac{r^2}{2}
\]

2. **Substitute and integrate with respect to \( r \)**:
\[
\int_{0}^{10} r \left(50 - \frac{r^2}{2}\right) \, dr = \int_{0}^{10} \left(50r - \frac{r^3}{2}\right) \, dr
\]
\[
\int_{0}^{10} 50r \, dr - \int_{0}^{10} \frac{r^3}{2} \, dr
\]

3. **Evaluate each term**:
\[
50 \int_{0}^{10} r \, dr = 50 \left[ \frac{r^2}{2} \right]_{0}^{10} = 50 \cdot \frac{100}{2} = 2500
\]
\[
\frac{1}{2} \int_{0}^{10} r^3 \, dr = \frac{1}{2} \left[ \frac{r^4}{4} \right]_{0}^{10} = \frac{1}{2} \cdot \frac{10000}{4} = \frac{5000}{4} = 1250
\]

4. **Combine the results**:
\[
2500 - 1250 = 1250
\]

5. **Integrate with respect to \( \theta \)**:
\[
\int_{0}^{2\pi} 1250 \, d\theta = 1250 \cdot 2\pi = 2500\pi
\]

### Final Answer:
\[
\text{Mass} = \pi \cdot 1250 = 1250\pi
\]

So, the mass is \(\boxed{1250\pi}\).

Would you like more details or have any questions about the steps?

### Further Questions for Understanding:
1. Why is it useful to convert this problem to cylindrical coordinates?
2. How is the volume element \( dV \) represented in cylindrical coordinates?
3. What is the significance of the density function \(\rho(x,y,z) = z\) in this problem?
4. How do you determine the integration limits when converting to cylindrical coordinates?
5. What are the general steps to evaluate a triple integral?
6. Why is it important to consider the order of integration in multiple integrals?
7. How does the geometry of the solid affect the limits of integration?
8. Can you explain the final step of integrating with respect to \(\theta\) and its result?

### Tip:
When dealing with triple integrals over solids bounded by planes and cones, converting to cylindrical coordinates often simplifies the process, especially when the solid exhibits circular symmetry.

Learn how to calculate the mass of a solid E positioned above a cone and below a plane using triple integrals in cylindrical coordinates. Detailed step-by-step solution provided.

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