Question

the dimensions of a rectangular shipping crate are 2.5 ft., 2.5 ft., and 4 ft. if there were a similar shipping crate with a scale factor of 3, what would be the volume of the new shipping crate? (1 point) \bigcirc 225 ft.\\(^3\\) \bigcirc 675 ft.\\(^3\\) \bigcirc 75 ft.\\(^3\\) \bigcirc 25 ft.\\(^3\\)

Explanation:

Step1: Find the volume of the original crate

The volume \( V \) of a rectangular prism is given by \( V = l \times w \times h \). For the original crate, \( l = 2.5 \) ft, \( w = 2.5 \) ft, and \( h = 4 \) ft. So, \( V_{\text{original}} = 2.5 \times 2.5 \times 4 \). Calculating that: \( 2.5 \times 2.5 = 6.25 \), and \( 6.25 \times 4 = 25 \) cubic feet.

Step2: Use the scale factor for volume

When a 3D figure is scaled by a scale factor \( k \), the volume scales by \( k^3 \). Here, \( k = 3 \), so the volume of the new crate \( V_{\text{new}} = V_{\text{original}} \times k^3 \). We know \( V_{\text{original}} = 25 \) and \( k^3 = 3^3 = 27 \)? Wait, no, wait. Wait, no, hold on. Wait, no, actually, wait: the scale factor for linear dimensions is 3, so volume scale factor is \( 3^3 = 27 \)? Wait, no, wait, no, wait. Wait, no, let's recalculate. Wait, original volume: \( 2.5 \times 2.5 \times 4 \). Let's compute that again. \( 2.5 \times 2.5 = 6.25 \), \( 6.25 \times 4 = 25 \). Then, scale factor for linear is 3, so new dimensions are \( 2.5 \times 3 \), \( 2.5 \times 3 \), \( 4 \times 3 \). So new length \( l' = 2.5 \times 3 = 7.5 \) ft, new width \( w' = 2.5 \times 3 = 7.5 \) ft, new height \( h' = 4 \times 3 = 12 \) ft. Then volume of new crate is \( 7.5 \times 7.5 \times 12 \). Let's calculate that: \( 7.5 \times 7.5 = 56.25 \), \( 56.25 \times 12 = 675 \) cubic feet. Alternatively, using the scale factor for volume: since volume scales with the cube of the linear scale factor, \( V_{\text{new}} = V_{\text{original}} \times 3^3 = 25 \times 27 = 675 \) cubic feet.

Answer:

\( 675 \text{ ft.}^3 \) (corresponding to the option "675 ft.³")