Question
Problem 14) Suppose we have a rectangle with a volume of 120 (pictured below). If we dilate this rectangle by a linear scale factor of k = 2, what would be the new volume?
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GPT-4o mini
To find the new volume of a rectangle after dilation, we can use the property of geometric scaling. When a three-dimensional shape (like a rectangular prism) is dilated by a linear scale factor \( k \), the volume scales by the factor \( k^3 \).
In this case, you have a rectangle (considered as a rectangular prism with a given volume of 120) and you dilate it by a scale factor \( k = 2 \).
1. Calculate the volume scaling factor:
\[
k^3 = 2^3 = 8
\]
2. Multiply the original volume by the volume scaling factor:
\[
\text{New Volume} = \text{Original Volume} \times k^3 = 120 \times 8 = 960
\]
Thus, after dilating the rectangle by a scale factor of \( k = 2 \), the new volume will be \( \boxed{960} \).
In this case, you have a rectangle (considered as a rectangular prism with a given volume of 120) and you dilate it by a scale factor \( k = 2 \).
1. Calculate the volume scaling factor:
\[
k^3 = 2^3 = 8
\]
2. Multiply the original volume by the volume scaling factor:
\[
\text{New Volume} = \text{Original Volume} \times k^3 = 120 \times 8 = 960
\]
Thus, after dilating the rectangle by a scale factor of \( k = 2 \), the new volume will be \( \boxed{960} \).
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