8. A cone has a volume of 300 units $$ ^{3} $$, and a cylinder has a volume of 100 units $$ ^{3} $$. Is it possible that the cylinder and the cone have the same height and the same radius? Why or why not?

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To determine if the cone and the cylinder can have the same height and radius, we need to compare their volume formulas.

1. **Volume of a Cone**: The formula for the volume $$ V_c $$ of a cone is given by:
   $$
   V_c = \frac{1}{3} \pi r^2 h
   $$
   where $$ r $$ is the radius and $$ h $$ is the height.

2. **Volume of a Cylinder**: The formula for the volume $$ V_{cy} $$ of a cylinder is given by:
   $$
   V_{cy} = \pi r^2 h
   $$

Given:
- Volume of the cone $$ V_c = 300 $$ units $$ ^{3} $$
- Volume of the cylinder $$ V_{cy} = 100 $$ units $$ ^{3} $$

Now, let's set up the equations based on the volumes:

For the cone:
$$
300 = \frac{1}{3} \pi r^2 h
$$

For the cylinder:
$$
100 = \pi r^2 h
$$

3. **Expressing Height in Terms of Radius**:
   - From the cone's volume equation:
   $$
   h = \frac{300 \cdot 3}{\pi r^2} = \frac{900}{\pi r^2}
   $$

- From the cylinder's volume equation:
   $$
   h = \frac{100}{\pi r^2}
   $$

4. **Setting the Heights Equal**:
   Since we want to find if they can have the same height, we set the two expressions for $$ h $$ equal to each other:
   $$
   \frac{900}{\pi r^2} = \frac{100}{\pi r^2}
   $$

5. **Simplifying the Equation**:
   We can cancel $$ \pi r^2 $$ from both sides (assuming $$ r \neq 0 $$):
   $$
   900 = 100
   $$

This is a contradiction, as 900 does not equal 100.

### Conclusion:
It is not possible for the cone and the cylinder to have the same height and radius because their volumes lead to a contradiction when equating their heights. The cone's volume is significantly larger than the cylinder's volume, which means they cannot share the same dimensions.
It's not possible for the cone and the cylinder to have the same height and radius because their volumes don't allow for equal heights with the same radius.

8. A cone has a volume of 300 units \( ^{3} \), and a cylinder has a volume of 100 units \( ^{3} \). Is it possible that the cylinder and the cone have the same height and the same radius? Why or why not?

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