Two cross sections of a right hexagonal pyramid are obtained by cutting the pyramid with planes parallel to the hexagonal base. The areas of the cross sections are $216\sqrt{3}$ square feet and $486\sqrt{3}$ square feet. The two planes are $8$ feet apart. How far from the apex of the pyramid is the larger cross section, in feet?
To find the distance from the apex to the larger cross section, I need to use the fact that the ratio of the areas of similar polygons is equal to the square of the ratio of their corresponding sides. So, if I let $s_1$ and $s_2$ be the side lengths of the smaller and larger hexagons, respectively, and $h_1$ and $h_2$ be their distances from the apex, then I have the equation $\frac{216\sqrt{3}}{486\sqrt{3}}=\frac{s_1^2}{s_2^2}=\frac{h_1^2}{h_2^2}$. Simplifying, I get $\frac{4}{9}=\frac{h_1^2}{h_2^2}$. Taking the square root of both sides, I get $\frac{2}{3}=\frac{h_1}{h_2}$. Now, I know that $h_1$ and $h_2$ differ by $8$ feet, so I can write $h_2=h_1+8$ and substitute that into the equation. I get $\frac{2}{3}=\frac{h_1}{h_1+8}$. Multiplying both sides by $h_1+8$, I get $2h_1+16=3h_1$. Subtracting $2h_1$ from both sides, I get $h_1=16$. Therefore, the distance from the apex to the smaller cross section is $16$ feet, and the distance from the apex to the larger cross section is $16+8=24$ feet.

