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:

\[ \boxed{1134} \]