Answer:
The larger cross section is 24 meters away from the apex.
Step-by-step explanation:
The cross section of a right hexagonal pyramid is a hexagon; therefore, let us first get some things clear about a hexagon.
The length of the side of the hexagon is equal to the radius of the circle that inscribes it.
The area is
[tex]A=\frac{3\sqrt{3} }{2} r^2[/tex]
Where [tex]r[/tex] is the radius of the inscribing circle (or the length of side of the hexagon).
Now we are given the areas of the two cross sections of the right hexagonal pyramid:[tex]A_1=216\:ft^2\: \:\:\:A_2=486\:ft^2[/tex]
From these areas we find the radius of the hexagons:
[tex]r_1=\sqrt{\frac{2A_1}{3\sqrt{3} } } =\sqrt{\frac{2*216}{3\sqrt{3} } }=\boxed{9.12ft}[/tex]
[tex]r_2=\sqrt{\frac{2A_2}{3\sqrt{3} } } =\sqrt{\frac{2*486}{3\sqrt{3} } }=\boxed{13.68ft}[/tex]
Now when we look at the right hexagonal pyramid from the sides ( as shown in the figure attached ), we see that [tex]r_1[/tex] [tex]r_2[/tex] form similar triangles with length [tex]H[/tex]
Therefore we have:
[tex]\frac{H-8}{r_1} =\frac{H}{r_2}[/tex]
We put in the numerical values of [tex]r_1[/tex], [tex]r_2[/tex] and solve for [tex]H[/tex]:
[tex]\boxed{H=\frac{8r_2}{r_2-r_1} =\frac{8*13.677}{13.68-9.12} =24\:feet.}[/tex]
