Question

4, 5 and 18. If the rectangular prism is dilated using a scale factor 1/2, What would be the volume of the dilated rectangular prism

Answer 1

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

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

For the original rectangular prism with dimensions 4, 5, and 18, the volume would be calculated as follows:

\[
\text{Volume}_{\text{original}} = 4 \times 5 \times 18
\]

Calculating that:

\[
\text{Volume}_{\text{original}} = 20 \times 18 = 360
\]

Now, if we dilate the rectangular prism using a scale factor of \( \frac{1}{2} \), the new dimensions will be:

\[
\text{New length} = 4 \times \frac{1}{2} = 2
\]
\[
\text{New width} = 5 \times \frac{1}{2} = 2.5
\]
\[
\text{New height} = 18 \times \frac{1}{2} = 9
\]

Now, we can find the volume of the dilated rectangular prism using the new dimensions:

\[
\text{Volume}_{\text{dilated}} = 2 \times 2.5 \times 9
\]

Calculating that:

\[
\text{Volume}_{\text{dilated}} = 5 \times 9 = 45
\]

Alternatively, you can use the property of volume in dilations, which states that the volume is affected by the cube of the scale factor. The scale factor is \( \frac{1}{2} \), so the volume will be multiplied by:

\[
\left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]

Therefore, the volume of the dilated rectangular prism can also be found by:

\[
\text{Volume}_{\text{dilated}} = \text{Volume}_{\text{original}} \times \frac{1}{8} = 360 \times \frac{1}{8} = 45
\]

Thus, the volume of the dilated rectangular prism is

\[
\boxed{45}
\]