Question

Problem 15) Suppose we have a triangular prism with volume of V = 42 (pictured below). If we dilate this figure by a linear scale factor of k=3, what would be the new volume?

Answer 1

To find the new volume of a triangular prism after dilation by a linear scale factor \( k \), we start with the relationship between volume and scale factor.

When a 3-dimensional figure is dilated by a scale factor \( k \), the new volume \( V' \) is given by the formula:

\[
V' = k^3 \times V
\]

where \( V \) is the original volume. In your case:

- The original volume \( V = 42 \)
- The linear scale factor \( k = 3 \)

Now, we can calculate the new volume:

\[
V' = 3^3 \times 42
\]

Calculating \( 3^3 \):

\[
3^3 = 27
\]

Now, substituting this back into the equation for the new volume:

\[
V' = 27 \times 42
\]

Now we perform the multiplication:

\[
V' = 1134
\]

Thus, the new volume of the triangular prism after dilation is:

\[
\boxed{1134}
\]