Two similar cylinders have volumes of 512π in.3

Question

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

Answer 1

To find the scale factor of the small cylinder to the larger cylinder based on their volumes, we can follow these steps:

1. Volume of the larger cylinder: \( V_1 = 512\pi , \text{in}^3 \)

2. Volume of the smaller cylinder: \( V_2 = 64\pi , \text{in}^3 \)

3. Calculate the ratio of their volumes: \[ \text{Scale Factor by Volume} = \frac{V_2}{V_1} = \frac{64\pi}{512\pi} = \frac{64}{512} = \frac{1}{8} \]

4. Finding the scale factor in terms of linear dimensions: Since the volumes of similar shapes are proportional to the cube of the scale factor of their corresponding dimensions, we take the cube root of the volume ratio to find the linear scale factor. \[ \text{Linear Scale Factor} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2} \]

Thus, the scale factor of the small cylinder to the larger cylinder in terms of linear dimensions is \(\frac{1}{2}\).