Question

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?

Answer 1

The larger cross section is 24 meters away from the apex.

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

`A=(3√(3) )/(2) r^2`

Where
`r` 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:
`A_1=216\:ft^2\: \:\:\:A_2=486\:ft^2`

From these areas we find the radius of the hexagons:

`r_1=\sqrt{(2A_1)/(3√(3) ) } =\sqrt{(2*216)/(3√(3) ) }=\boxed{9.12ft}`

`r_2=\sqrt{(2A_2)/(3√(3) ) } =\sqrt{(2*486)/(3√(3) ) }=\boxed{13.68ft}`

Now when we look at the right hexagonal pyramid from the sides ( as shown in the figure attached ), we see that
`r_1`
`r_2` form similar triangles with length
`H`

Therefore we have:

`(H-8)/(r_1) =(H)/(r_2)`

We put in the numerical values of
`r_1`,
`r_2` and solve for
`H`:

`\boxed{H=(8r_2)/(r_2-r_1) =(8*13.677)/(13.68-9.12) =24\:feet.}`