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 a cone and a cylinder can have the same height and radius while having different volumes, we can use the formulas for the volumes of a cone and a cylinder.

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

2. **Volume of a cylinder**: 
   $$
   V_{cy} = \pi r^2 h
   $$
   where $$ V_{cy} $$ is the volume of the cylinder.

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

Now, if both the cone and the cylinder have the same height $$ h $$ and radius $$ r $$, we can set up the equations based on their volumes:

From the cone's volume:
$$
300 = \frac{1}{3} \pi r^2 h
$$

From the cylinder's volume:
$$
100 = \pi r^2 h
$$

Now, we can express $$ h $$ in terms of $$ r $$ from both equations.

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

Substituting this expression for $$ h $$ into the cone's volume equation:
$$
300 = \frac{1}{3} \pi r^2 \left(\frac{100}{\pi r^2}\right)
$$

Now, simplifying this:
$$
300 = \frac{100}{3}
$$

This equation does not hold true, as $$ 300 \neq \frac{100}{3} $$.

Thus, it is not possible for the cone and the cylinder to have the same height and radius while having the given volumes. The volumes are fundamentally different due to the geometric properties of the shapes.

In conclusion, the answer is **no**, it is not possible for the cylinder and the cone to have the same height and radius given their volumes.
It's not possible for the cone and the cylinder to have the same height and radius with the given volumes because the equations derived from their volume formulas don't match.

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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