Find the weight of the heaviest circular cylinder can be cut from a 16-lb shot.
 

To find the weight of the heaviest circular cylinder that can be cut from a 16-pound shot (a sphere), we can follow these steps:

### 1. Volume of the Sphere
First, calculate the volume of the 16-pound spherical shot. Since the shot is a sphere, we can use the formula for the volume of a sphere:

\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]

We are not given the radius \( r \) directly, but we know the weight of the sphere is proportional to its volume. Let's denote the density of the material as \( \rho \). Then:

\[
\text{Weight of the sphere} = \rho \times V_{\text{sphere}} = 16 \, \text{lb}
\]

### 2. Volume of the Cylinder
Next, the volume of the cylinder that can be cut from this sphere can be given by:

\[
V_{\text{cylinder}} = \pi r^2 h
\]

Here, \( r \) is the radius of the cylinder, and \( h \) is its height. The largest cylinder that can be cut out of a sphere will have a diameter equal to the diameter of the sphere, so \( h = 2r \) (where \( r \) is the radius of the sphere).

Substituting this, we get:

\[
V_{\text{cylinder}} = \pi r^2 (2r) = 2 \pi r^3
\]

### 3. Weight of the Cylinder
Since the material is uniform, the weight of the cylinder will also be proportional to its volume. Let's denote the weight of the cylinder as \( W_{\text{cylinder}} \):

\[
W_{\text{cylinder}} = \rho \times V_{\text{cylinder}} = \rho \times 2 \pi r^3
\]

### 4. Ratio of Cylinder to Sphere Volume
To find the maximum weight, we need to compare the volumes of the cylinder and sphere:

\[
\frac{W_{\text{cylinder}}}{W_{\text{sphere}}} = \frac{\rho \times 2 \pi r^3}{\rho \times \frac{4}{3} \pi r^3} = \frac{2 \pi r^3}{\frac{4}{3} \pi r^3} = \frac{2}{\frac{4}{3}} = \frac{3}{2}
\]

Therefore:

\[
W_{\text{cylinder}} = \frac{3}{2} \times W_{\text{sphere}} = \frac{3}{2} \times 16 \, \text{lb} = 12 \, \text{lb}
\]

### Conclusion
The weight of the heaviest cylinder that can be cut from a 16-pound shot is 12 pounds.

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Would you like more details, or do you have any questions?

Here are 8 follow-up questions:
1. How is the ratio between the cylinder and sphere volumes derived?
2. Can this method be applied to any other shapes?
3. What if the cylinder's height is not equal to the diameter of the sphere?
4. How does the material density affect the weight calculation?
5. What if the sphere had a different weight, say 20 lb?
6. Could we consider cutting other shapes from the sphere? 
7. How would the result change if the cylinder were hollow?
8. Could this problem be solved using calculus?

**Tip:** Always check if the shape you're working with can be inscribed within another for maximum efficiency.

Learn how to calculate the weight of the heaviest cylinder that can be cut from a 16-pound spherical shot. This problem involves calculating volumes, understanding density, and applying geometric principles. Suitable for advanced high school students.

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