Question

if the above pyramid 12ft. 6ft. 15ft. is dilated using a scale factor of 2 what would be the new volume? to find the volume of a pyramid, us V = l w h/3

Answer 1

To find the volume of the pyramid after dilation by a scale factor of 2, we first need to calculate the volume of the original pyramid and then determine the volume of the dilated pyramid.

1. **Calculate the volume of the original pyramid:**
The formula for the volume of a pyramid is given by:
\[
V = \frac{l \times w \times h}{3}
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

From the dimensions provided:
- Length (\( l \)) = 12 ft
- Width (\( w \)) = 6 ft
- Height (\( h \)) = 15 ft

Substitute the dimensions into the volume formula:
\[
V = \frac{12 \times 6 \times 15}{3}
\]

First, calculate \( 12 \times 6 \times 15 \):
\[
12 \times 6 = 72
\]
\[
72 \times 15 = 1080
\]

Now plug this back into the volume formula:
\[
V = \frac{1080}{3} = 360 \text{ cubic feet}
\]

2. **Calculate the new dimensions after dilation:**
Since we are dilating the pyramid by a scale factor of 2, we multiply each dimension by 2:
- New length (\( l' \)) = \( 12 \times 2 = 24 \) ft
- New width (\( w' \)) = \( 6 \times 2 = 12 \) ft
- New height (\( h' \)) = \( 15 \times 2 = 30 \) ft

3. **Calculate the volume of the dilated pyramid:**
Now we use the new dimensions to calculate the new volume \( V' \):
\[
V' = \frac{l' \times w' \times h'}{3} = \frac{24 \times 12 \times 30}{3}
\]

Calculate \( 24 \times 12 \times 30 \):
\[
24 \times 12 = 288
\]
\[
288 \times 30 = 8640
\]

Now plug this back into the volume formula:
\[
V' = \frac{8640}{3} = 2880 \text{ cubic feet}
\]

4. **Conclusion:**
The new volume of the pyramid after dilation is:
\[
\boxed{2880} \text{ cubic feet}
\]