Question

the pyramid is dilated using a scale factor of 2, what would be the new volume? to vide the volume of a pyramid, use V = lwh/3
Length = 12ft
width = 6ft
height = 15 ft

Answer 1

To find the new volume of the pyramid after it has been dilated with a scale factor of 2, we first need to find the original volume.

The volume \( V \) of a pyramid is given by the formula:

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

where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.

### Step 1: Calculate the original volume

Given:
- Length \( l = 12 \, \text{ft} \)
- Width \( w = 6 \, \text{ft} \)
- Height \( h = 15 \, \text{ft} \)

Plugging these values into the formula:

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

Calculate the product:

\[
12 \times 6 = 72
\]

Then,

\[
72 \times 15 = 1080
\]

Now, divide by 3:

\[
V = \frac{1080}{3} = 360 \, \text{ft}^3
\]

### Step 2: Calculate the new volume after dilation

When a three-dimensional shape is dilated by a scale factor \( k \), its volume is scaled by a factor of \( k^3 \). In this case, \( k = 2 \), so:

\[
\text{Scale factor for volume} = k^3 = 2^3 = 8
\]

Now, multiply the original volume by this scale factor:

\[
\text{New Volume} = 360 \times 8 = 2880 \, \text{ft}^3
\]

### Conclusion

After dilating the pyramid using a scale factor of 2, the new volume is:

\[
\boxed{2880 \, \text{ft}^3}
\]