Question

15 high
12 wide
6 long
If the above pyramid is dilated using a scale factor of 2, what would be the new volume? To find the volume of a pyramid, use V = l w h3 (1 point) ft.3

Answer 1

To find the volume of a pyramid, the formula is:

\[
V = \frac{1}{3} \times l \times w \times h
\]

where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the pyramid.

Given your initial dimensions:
- Height (\( h \)) = 15 ft
- Width (\( w \)) = 12 ft
- Length (\( l \)) = 6 ft

First, calculate the original volume:

\[
V = \frac{1}{3} \times 6 \times 12 \times 15
\]

Calculating this step-by-step:

\[
6 \times 12 = 72
\]
\[
72 \times 15 = 1080
\]
\[
V = \frac{1}{3} \times 1080 = 360 \text{ ft}^3
\]

Now, if the pyramid is dilated using a scale factor of 2, all dimensions will be multiplied by this scale factor. The new dimensions will be:
- New Height = \( 15 \times 2 = 30 \) ft
- New Width = \( 12 \times 2 = 24 \) ft
- New Length = \( 6 \times 2 = 12 \) ft

Next, calculate the new volume using the new dimensions:

\[
V' = \frac{1}{3} \times 12 \times 24 \times 30
\]

Calculating this step-by-step:

\[
12 \times 24 = 288
\]
\[
288 \times 30 = 8640
\]
\[
V' = \frac{1}{3} \times 8640 = 2880 \text{ ft}^3
\]

So, the new volume after the dilation is \( \text{2880 ft}^3 \).