Question
What is the ratio of corresponding linear dimensions for two similar boxes, where one box has a volume 125 times that of the other?
Answers
GPT-4o mini
To find the ratio of corresponding linear dimensions for two similar boxes where one box has a volume 125 times that of the other, we begin by recalling that the volumes of similar shapes scale with the cube of the ratio of their corresponding linear dimensions.
Let the ratio of the corresponding linear dimensions be \( r \). According to the properties of similar figures, their volumes relate to the ratio of their linear dimensions as follows:
\[
\text{Volume Ratio} = r^3
\]
In this case, we know that one box has a volume that is 125 times that of the other. Therefore:
\[
\frac{V_2}{V_1} = 125
\]
This gives us the equation:
\[
r^3 = 125
\]
To find \( r \), we take the cube root of both sides:
\[
r = \sqrt[3]{125}
\]
Since \( 125 = 5^3 \), we get:
\[
r = \sqrt[3]{5^3} = 5
\]
Thus, the ratio of the corresponding linear dimensions of the two boxes is:
\[
\boxed{5}
\]
Let the ratio of the corresponding linear dimensions be \( r \). According to the properties of similar figures, their volumes relate to the ratio of their linear dimensions as follows:
\[
\text{Volume Ratio} = r^3
\]
In this case, we know that one box has a volume that is 125 times that of the other. Therefore:
\[
\frac{V_2}{V_1} = 125
\]
This gives us the equation:
\[
r^3 = 125
\]
To find \( r \), we take the cube root of both sides:
\[
r = \sqrt[3]{125}
\]
Since \( 125 = 5^3 \), we get:
\[
r = \sqrt[3]{5^3} = 5
\]
Thus, the ratio of the corresponding linear dimensions of the two boxes is:
\[
\boxed{5}
\]