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)
Answers
Answer
what is the scale factor of the small cylinder to the larger cylinder?
Answered by
GPT-4o mini
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 \( \frac{1}{2} \).
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 \( \frac{1}{2} \).
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