Answer:
The height of the pyramid is [tex]27.56\ ft[/tex]
Step-by-step explanation:
we know that
The volume of the pyramid is equal to
[tex]V=\frac{1}{3}BH[/tex]
where
B is the area of the base of the pyramid
H is the height of the pyramid
step 1
Find the area B of the regular hexagonal base
we know that
The perimeter of a regular hexagon is
[tex]P=6b[/tex]
where
b is the length side of the hexagon
we have
[tex]P=84\ ft[/tex]
substitute
[tex]84=6b[/tex]
solve for b
[tex]b=14\ ft[/tex]
Remember that the area of a regular hexagon is the same that the area of six equilateral triangles
Determine the area of the six equilateral triangles, applying the formula of the law of sines
[tex]B=6[\frac{1}{2}b^2sin(60\°)][/tex]
substitute the value of b
[tex]B=6[\frac{1}{2}(14)^2sin(60\°)][/tex]
[tex]B=509.22\ ft^2[/tex]
step 2
Find the height of the pyramid
[tex]V=\frac{1}{3}BH[/tex]
we have
[tex]V=4,677.85\ ft^3[/tex]
[tex]B=509.22\ ft^2[/tex]
substitute
[tex]4,677.85=\frac{1}{3}(509.22)H[/tex]
solve for H
[tex]14,033.55=(509.22)H[/tex]
[tex]H=14,033.55/(509.22)H[/tex]
[tex]H=27.56\ ft[/tex]
