Question

imagine that a rectangular prism similar to the one shown has a scale factor of \\(\frac{1}{2}\\). what is the volume of the new rectangular prism?\
(1 point)\
\\(\bigcirc\\) 120 \\(\text{cm}^3\\)\
\\(\bigcirc\\) 60 \\(\text{cm}^3\\)\
\\(\bigcirc\\) 30 \\(\text{cm}^3\\)\
\\(\bigcirc\\) 15 \\(\text{cm}^3\\)

Explanation:

Step1: Find the volume of the original rectangular prism

The formula for the volume of a rectangular prism is \( V = l \times w \times h \). From the diagram, the length \( l = 5\) cm, width \( w = 4\) cm, and height \( h = 6\) cm. So the volume of the original prism is \( V_{original}=5\times4\times6 = 120\) \( cm^3 \).

Step2: Determine the scale factor for the volume

When a three - dimensional figure is scaled by a scale factor \( k \), the volume of the new figure is \( V_{new}=k^{3}\times V_{original} \). Here, the scale factor \( k=\frac{1}{2} \). So the scale factor for the volume is \( k^{3}=(\frac{1}{2})^{3}=\frac{1}{8} \).

Step3: Calculate the volume of the new rectangular prism

We know \( V_{original} = 120\) \( cm^3 \) and the scale factor for volume is \( \frac{1}{8} \). Then \( V_{new}=120\times\frac{1}{8}=15 \) \( cm^3 \).

Answer:

\( 15\ cm^3 \) (corresponding to the option "15 \( cm^3 \)")