Question

4,3 and 2. Determine the volume of the rectangular prism if it were dilated using a scale factor of 4.

Answer 1

To determine the volume of a rectangular prism, you can use the formula:

\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]

In this case, the dimensions of the rectangular prism are given as 4, 3, and 2. Thus, the initial volume of the prism can be calculated as follows:

\[
\text{Volume} = 4 \times 3 \times 2 = 24
\]

Now, if the prism is dilated using a scale factor of 4, each dimension will be multiplied by that scale factor. Therefore, the new dimensions will be:

\[
\text{New length} = 4 \times 4 = 16
\]
\[
\text{New width} = 3 \times 4 = 12
\]
\[
\text{New height} = 2 \times 4 = 8
\]

Now, calculate the volume of the dilated prism:

\[
\text{New Volume} = \text{New length} \times \text{New width} \times \text{New height}
\]
\[
\text{New Volume} = 16 \times 12 \times 8
\]

Calculating step by step:

1. \( 16 \times 12 = 192 \)
2. \( 192 \times 8 = 1536 \)

Thus, the volume of the rectangular prism after dilation is:

\[
\text{New Volume} = 1536
\]

Alternatively, there is a shortcut for calculating the volume change due to dilation. The volume of a shape scales by the cube of the scale factor. Since the scale factor is 4, the volume increases by:

\[
\text{New Volume} = \text{Old Volume} \times (4^3) = 24 \times 64 = 1536
\]

Either way, the final volume of the dilated rectangular prism is:

\[
\boxed{1536}
\]