Mathematics: the base of a regular pyramid is a hexagon whose perimeter is 84 feet. the volume of the pyramid is approximately 4,677.85 cubic feet. find the height of the pyramid.

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the base of a regular pyramid is a hexagon whose

Ответ:

MrKrinkle77

11.01.2022

Mathematics

The height of the pyramid is $27.56\ ft$

Step-by-step explanation:

we know that

The volume of the pyramid is equal to

$V=\frac{1}{3}BH$

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

P=6b

where

b is the length side of the hexagon

we have

$P=84\ ft$

substitute

84=6b

solve for b

$b=14\ ft$

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

$B=6[\frac{1}{2}b^2sin(60\°)]$

substitute the value of b

$B=6[\frac{1}{2}(14)^2sin(60\°)]$

$B=509.22\ ft^2$

step 2

Find the height of the pyramid

$V=\frac{1}{3}BH$

we have

$V=4,677.85\ ft^3$

$B=509.22\ ft^2$

substitute

$4,677.85=\frac{1}{3}(509.22)H$

solve for H

14,033.55=(509.22)H

H=14,033.55/(509.22)H

$H=27.56\ ft$

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the base of a regular pyramid is a hexagon whose perimeter is 84 feet. the volume of the pyramid is approximately 4,677.85 cubic feet. find the height of the pyramid.

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