Question

Use the image to answer the question.

A 4-sided pyramid with height from the top to base labeled 15 feet, the length of the base labeled 12 feet, and the width of the base labeled 6 feet.

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 = lwh/3
(1 point)
ft.3

Answer 1

To find the volume of the original pyramid, we will first use the formula for the volume of a pyramid:

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

From the information given:
- Height (h) = 15 feet
- Length (l) = 12 feet
- Width (w) = 6 feet

Now we can calculate the volume of the original pyramid:

\[
V = \frac{12 \times 6 \times 15}{3}
\]
\[
V = \frac{720}{3}
\]
\[
V = 240 \text{ ft}^3
\]

Now, when we dilate the pyramid with a scale factor of 2, all dimensions of the pyramid will be multiplied by 2. Therefore:
- New height = \(15 \times 2 = 30\) feet
- New length = \(12 \times 2 = 24\) feet
- New width = \(6 \times 2 = 12\) feet

Now we can recalculate the volume of the dilated pyramid:

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

Where:
- New height (h') = 30 feet
- New length (l') = 24 feet
- New width (w') = 12 feet

Calculating the volume of the dilated pyramid:

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

Thus, the new volume of the pyramid after dilation is:

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