Apothem Of A Prism Find The Length Of The Apothem Of A Hexagonal Prism If The Volume Is 120sqrt3 Cubic In. And The Height Is 5 In
Find the length of the apothem of a hexagonal prism if the volume is 120sqrt3 cubic in. and the height is 5 in - apothem of a prism
Let x = side of the hexagon
a = x √ 3 / 2
Perimeter (P) = 6x
Area of hexagon = pa / 2
volume = pa / 2
120 √ 3 = 6x * x √ 3 / 2 * 5 / 2
120 = 3x ²
40 = x ²
x = 2 √ 10
a = 2 √ 10 √ 3 / 2
= √ 30
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1 comments:
Volume = 1 / 3 · Area_base · Height
-> Area_base = 3 • Volume / height = 3.120 × √ 3 / 5 = 72 × √ 3 in ²
Area_base = 6 • 1 / 2 • RL · with
tan60 ° = R / (L / 2) -> L = 2 • r/tan60 º = 2 • R / √ 3 = 2 √ 3 / 3 • R
-> 72 × √ 3 = 6 • 1 / 2 • 2 √ 3 / 3 • R ² = 2 • √ 3 • R ²
R ² = 36 -> r = 6 = apothem
Sludos.
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