To solve this problem, let's first understand the relationship between the side lengths, areas, and volumes of similar figures. When two figures are similar, their corresponding side lengths have a common ratio. In this case, the side length of the smaller pyramid is [tex]\(\frac{3}{4}\)[/tex] of the side length of the larger pyramid.
Here's the step-by-step solution:
1. Identify the ratio of the side lengths:
The side length ratio between the smaller pyramid and the larger pyramid is given as [tex]\(\frac{3}{4}\)[/tex].
2. Relationship between areas of similar figures:
The ratio of the areas of two similar figures is the square of the ratio of their corresponding side lengths. This is because area is a two-dimensional measure and thus involves squaring the linear dimensions.
Let’s denote the side length ratio as [tex]\( \frac{a}{b} = \frac{3}{4} \)[/tex].
Therefore, the ratio of the base areas of the two pyramids will be:
[tex]\[
\left(\frac{3}{4}\right)^2
\][/tex]
3. Calculate the area ratio:
[tex]\[
\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}
\][/tex]
So, the fraction that represents the ratio of the base area of the smaller pyramid to the base area of the larger pyramid is:
[tex]\(\frac{9}{16}\)[/tex].
