SOLUTION: The frustum has regular hexagon bases. The upper base measures 16 ft. on a side and the lower base measures 29 ft. on a side. The altitude of the frustum is 16 ft. Find the mass o
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Question 1111892: The frustum has regular hexagon bases. The upper base measures 16 ft. on a side and the lower base measures 29 ft. on a side. The altitude of the frustum is 16 ft. Find the mass of the frustum, if its density is 99 lbs. per cu. ft.
Found 2 solutions by Alan3354, greenestamps:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
The frustum has regular hexagon bases. The upper base measures 16 ft. on a side and the lower base measures 29 ft. on a side. The altitude of the frustum is 16 ft. Find the mass of the frustum, if its density is 99 lbs. per cu. ft.
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Find the volume.
The side length of the hexagon between the bases (8 ft from either) is (29+15)/2 = 22.5 ft
The area is ns^2*cot(180/n)/4 where n = # of sides & s = side length
Area = 6*22.5^2*sqrt(3)/4 = 759.375*sqrt(3) sq ft
Vol = Area*16 = 12150*sqrt(3) cubic ft
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Mass = Vol*density
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
The answer from the other tutor is not correct. With a frustum (truncated pyramid), you can't just use the length of a side halfway between the two bases to find the volume of the frustum....
Here are three ways to find the volume of the frustum and thus find its weight.
(1) Consider the frustum as a truncated hexagonal pyramid; find the volume of the original pyramid and subtract off the volume of the pyramid that was cut off.
Use similar triangles to find the height h of the original pyramid:
The volume of the original pyramid is one-third the area of the base, times the height; the area of the regular hexagonal base is the area of 6 equilateral triangles with side length 29:
The volume of the pyramid that is cut off is one-third the area of the base, times the height:
The volume of the frustum is the difference between the volumes of the two pyramids:
(2) In that first method, we found the volume of the pyramid that was cut off directly, using the formula for the volume of a pyramid. We can also find the volume of that pyramid using the fact that the original pyramid and the pyramid that was cut off are similar figures.
The scale factor between the two pyramids is 16:29; the ratio of the volumes of the two pyramids is (16:29)^3.
So the volume of the frustum can be calculated as the volume of the original pyramid, multiplied by (1 - (16/29)^3):
(3) There is a rather obscure formula for the volume of a frustum of a pyramid with a regular polygonal base; for a frustum with a regular hexagonal base, with base side lengths 29 and 16 and height 16, the formula is
Evaluating this expression of course gives the same answer of 21629.85 for the volume of the frustum.
That answer is in cubic feet; then to find the weight you simply multiply that by the density in pounds per cubic foot:
2,141,355 pounds -- a rather heavy chunk of something.
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