Question

You are comparing two cylindrical containers. Cylinder 1 has a volume of
and a base radius of 6. Cylinder 2 has a volume of
and a radius of 10. Which container is taller? How do you know?

Answer 1

To compare the heights of the two cylindrical containers, we will need to use the formula for the volume of a cylinder, which is:

\[
V = \pi r^2 h
\]

where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.

### For Cylinder 1:
- **Volume**: \( V_1 \)
- **Base Radius**: \( r_1 = 6 \)

Using the volume formula, we can express the height \( h_1 \) of Cylinder 1 as:

\[
h_1 = \frac{V_1}{\pi r_1^2} = \frac{V_1}{\pi (6^2)} = \frac{V_1}{36\pi}
\]

### For Cylinder 2:
- **Volume**: \( V_2 \)
- **Base Radius**: \( r_2 = 10 \)

Similarly, the height \( h_2 \) of Cylinder 2 can be expressed as:

\[
h_2 = \frac{V_2}{\pi r_2^2} = \frac{V_2}{\pi (10^2)} = \frac{V_2}{100\pi}
\]

### Comparison of Heights:
To determine which container is taller, we need to compare \( h_1 \) and \( h_2 \):

\[
h_1 = \frac{V_1}{36\pi} \quad \text{and} \quad h_2 = \frac{V_2}{100\pi}
\]

This can be simplified to comparing:

\[
h_1 \cdot 36\pi \quad \text{to} \quad h_2 \cdot 100\pi
\]

So we are effectively comparing:

\[
V_1 \cdot 100 \quad \text{to} \quad V_2 \cdot 36
\]

### Conclusion:
To determine which cylinder is taller, we need specific values for \( V_1 \) and \( V_2 \). If:

- \( V_1 \cdot 100 > V_2 \cdot 36 \), then Cylinder 1 is taller.
- \( V_1 \cdot 100 < V_2 \cdot 36 \), then Cylinder 2 is taller.

If you provide the values of \( V_1 \) and \( V_2 \), we can perform the actual comparison.

Answer 2

answer as a 7 year old