Alright, let's solve this problem step by step.
1. Understand the given data:
- The volume of the larger pyramid: [tex]\( V_{\text{large}} = 729 \, \text{cm}^3 \)[/tex]
- The volume of the smaller pyramid: [tex]\( V_{\text{small}} = 64 \, \text{cm}^3 \)[/tex]
- The perimeter of the base of the smaller pyramid: [tex]\( P_{\text{small}} = 8 \, \text{cm} \)[/tex]
2. Determine the ratio of the volumes:
The ratio of the volumes of the similar pyramids can be found by dividing the volume of the larger pyramid by the volume of the smaller pyramid:
[tex]\[
\text{Ratio of volumes} = \frac{V_{\text{large}}}{V_{\text{small}}} = \frac{729}{64} = 11.390625
\][/tex]
3. Find the ratio of the linear dimensions:
Since the pyramids are similar, the ratio of the linear dimensions (such as the perimeters of the bases) is the cube root of the ratio of the volumes. Therefore, we need to find the cube root of [tex]\( 11.390625 \)[/tex]:
[tex]\[
\text{Ratio of linear dimensions} = \sqrt[3]{11.390625} = 2.25
\][/tex]
4. Calculate the perimeter of the base of the larger pyramid:
The perimeter of the base of the larger pyramid is the perimeter of the base of the smaller pyramid multiplied by the ratio of the linear dimensions:
[tex]\[
P_{\text{large}} = P_{\text{small}} \times \text{Ratio of linear dimensions} = 8 \times 2.25 = 18 \, \text{cm}
\][/tex]
So, the perimeter of the base of the larger pyramid is [tex]\( 18 \, \text{cm} \)[/tex].
Final Answer:
[tex]\( 18 \, \text{cm} \)[/tex]
