Imagine that a rectangular prism similar to the one shown has a scale factor of `\frac{1}{2}`.

Question

What is the volume of the new rectangular prism?

GPT-4o mini · Answer

When scaling the dimensions of a rectangular prism by a scale factor \( k \), the volume of the new prism is scaled by \( k^3 \).

Given that the scale factor is \( \frac{1}{2} \), we can calculate the volume scaling factor as follows:

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

If we want to find the volume of the new rectangular prism, we need to know the volume of the original prism. Let's denote the volume of the original rectangular prism as \( V \).

The volume of the new rectangular prism can be calculated with the formula:

\[ V_{\text{new}} = V \times \frac{1}{8} \]

So if you can provide the volume \( V \) of the original prism, I can give you the exact volume of the new prism. If volume \( V \) is not provided, the new volume will be \( \frac{1}{8} \) times that of the original prism.

Answer

w=4,l=6,h=3

GPT-4o mini · Answer

To find the volume of the original rectangular prism, we can use the formula for the volume of a rectangular prism, which is:

\[ V = w \times l \times h \]

Given the dimensions:

Width \( w = 4 \)
Length \( l = 6 \)
Height \( h = 3 \)

We can calculate the volume \( V \):

\[ V = 4 \times 6 \times 3 \]

Calculating this step-by-step:

\[ V = 4 \times 6 = 24 \] \[ V = 24 \times 3 = 72 \]

So, the volume of the original rectangular prism is \( 72 \) cubic units.

Now, applying the scale factor of \( \frac{1}{2} \) to find the volume of the new rectangular prism:

\[ V_{\text{new}} = V \times \frac{1}{8} = 72 \times \frac{1}{8} \]

Calculating \( V_{\text{new}} \):

\[ V_{\text{new}} = 72 \div 8 = 9 \]

Thus, the volume of the new rectangular prism is \( \boxed{9} \) cubic units.