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To solve this problem, we need to optimize the dimensions of a cone that’s formed by cutting a sector from a circular sheet with radius 23 and then welding the sheet's edges to create the cone.

### Steps to Solve:

1. **Understand the Geometry of the Cone**:
   - The radius of the original circular sheet is given as \( R = 23 \).
   - When the sector is cut and folded, the arc length of the sector becomes the circumference of the cone’s base.

2. **Formulas to Use**:
   - Let \( r \) be the radius of the cone's base, and \( h \) be its height.
   - The slant height of the cone, which is the same as the radius of the original circular sheet, is 23. Thus, \( s = 23 \).
   - The arc length of the sector, which becomes the circumference of the cone’s base, is \( 2\pi r \).

3. **Setting Up Relationships**:
   - By the Pythagorean theorem for the cone, we know \( s^2 = r^2 + h^2 \).
   - Plugging \( s = 23 \), we get:
     \[
     23^2 = r^2 + h^2 \Rightarrow 529 = r^2 + h^2.
     \]
   - Solving for \( h \) in terms of \( r \):
     \[
     h = \sqrt{529 - r^2}.
     \]

4. **Volume of the Cone**:
   - The volume \( V \) of a cone is given by \( V = \frac{1}{3} \pi r^2 h \).
   - Substituting \( h \) from the previous step:
     \[
     V = \frac{1}{3} \pi r^2 \sqrt{529 - r^2}.
     \]

5. **Maximizing the Volume**:
   - To find the value of \( r \) that maximizes \( V \), take the derivative of \( V \) with respect to \( r \) and set it to zero. Solve this equation to find the optimal value of \( r \), then use it to find \( h \).

After calculating, we find the exact values:

- **Radius (r)**: \( \frac{23}{\sqrt{3}} \)
- **Height (h)**: \( \frac{23\sqrt{2}}{\sqrt{3}} \)

Let me know if you need detailed steps for finding these values or have any questions. 

### Further Questions to Consider:
1. How is the relationship between the slant height and the cone dimensions used here?
2. Can this approach be applied to different starting sheet radii?
3. What happens to the cone's volume if a smaller sector is cut?
4. How would the formula change if the cone had a specific height restriction?
5. Why does the maximum volume occur at these specific values of \( r \) and \( h \)?

**Tip**: Always check if derivatives are set up correctly when optimizing geometric shapes!

A cone is constructed by cutting a sector from a circular sheet of metal with radius 23. The cut sheet is then folded up and welded. Find the radius and height of the cone with maximum volume that can be formed in this way.

Explore the solution to find the maximum volume of a cone created by cutting a sector from a circular sheet with radius 23 and welding it. This problem involves key geometric concepts such as cone volume, optimization, and using the Pythagorean Theorem. Suitable for advanced high school math students.

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