To solve the problem of finding the perimeter of the base of the larger pyramid given that the two triangular pyramids are similar, follow these steps:
1. Identify the given data:
- Volume of the larger pyramid: [tex]\( 729 \, cm^3 \)[/tex]
- Volume of the smaller pyramid: [tex]\( 64 \, cm^3 \)[/tex]
- Perimeter of the base of the smaller pyramid: [tex]\( 8 \, cm \)[/tex]
2. Determine the scale factor based on volumes:
- Since the pyramids are similar, the volumes of similar figures are proportional to the cube of their scale factor.
- Let the scale factor be [tex]\( k \)[/tex].
- Then, [tex]\( \left(\frac{\text{volume of the larger pyramid}}{\text{volume of the smaller pyramid}}\right) = k^3 \)[/tex].
Therefore:
[tex]\[
k^3 = \left(\frac{729}{64}\right)
\][/tex]
3. Calculate the scale factor [tex]\( k \)[/tex]:
- We find the cube root of the ratio of the volumes:
[tex]\[
k = \left(\frac{729}{64}\right)^{\frac{1}{3}}
\][/tex]
[tex]\[
k = 2.25
\][/tex]
4. Calculate the perimeter of the base of the larger pyramid:
- Since the pyramids are similar, the perimeter of the bases are proportional to the linear scale factor [tex]\( k \)[/tex].
- Hence:
[tex]\[
\text{Perimeter of base of larger pyramid} = \text{Perimeter of base of smaller pyramid} \times k
\][/tex]
[tex]\[
\text{Perimeter of base of larger pyramid} = 8 \, cm \times 2.25
\][/tex]
Therefore:
[tex]\[
\text{Perimeter of base of larger pyramid} = 18 \, cm
\][/tex]
Thus, the perimeter of the base of the larger pyramid is [tex]\( 18 \, cm \)[/tex].
