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?

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

Where 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:

From these areas we find the radius of the hexagons:

Now when we look at the right hexagonal pyramid from the sides ( as shown in the figure attached ), we see that form similar triangles with length

Therefore we have:

We put in the numerical values of , and solve for :

step-by-step explanation: