Question

Two similar cylinders have volumes of 512π in.3 and 64π in.3 . By comparing volume, what is the scale factor of the small cylinder to the larger cylinder?(1 point)

Answer 1

what is the scale factor of the small cylinder to the larger cylinder?

Answer 2

To find the scale factor between the two similar cylinders based on their volumes, we use the relationship between the volumes of similar three-dimensional figures.

The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

For two similar cylinders, the volumes compare as follows:
\[
\frac{V_1}{V_2} = k^3
\]
where \( k \) is the scale factor between the two cylinders.

Given:
- Volume of the larger cylinder \( V_1 = 512\pi \, \text{in}^3 \)
- Volume of the smaller cylinder \( V_2 = 64\pi \, \text{in}^3 \)

First, we can simplify the volume ratio:
\[
\frac{V_1}{V_2} = \frac{512\pi}{64\pi} = \frac{512}{64} = 8
\]

Now, we set up the equation:
\[
k^3 = 8
\]

To find \( k \), we take the cube root of both sides:
\[
k = \sqrt[3]{8} = 2
\]

Thus, the scale factor of the small cylinder to the larger cylinder is:
\[
\frac{1}{k} = \frac{1}{2}
\]

Therefore, the scale factor of the small cylinder to the larger cylinder is \( \frac{1}{2} \).