SOLUTION: The flowerpot has a square base 12 in. at the bottom and 18 in. at the top has a uniform thickness of 3 in. and is 1ft. high. find the amount of material necessary to construct 100

Question 1058177: The flowerpot has a square base 12 in. at the bottom and 18 in. at the top has a uniform thickness of 3 in. and is 1ft. high. find the amount of material necessary to construct 1000 such pots. (Neglect the drain in the bottom) illustration pls
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The drawing represents a pyramid,
with base side 18" and height 36",
cut along the red lines
at 12" and 24" above its base.
There is also a green cut, 3" below the lower red cut.
Those red cuts are 6" and 12" wide.
If we remove the part above the lower red cut,
it looks like your pot,
except that you need to dig out the inside,
along the inner black and green lines.
We have a lot of pyramids there.
The small one above the top red line has
a 6" base side and a 12" height.
Its volume, in cubic inches, is
.
The other pyramids are all similar,
scaled up versions of that one.
Luckily, when you scale up by a factor of ,
volume increases by a factor of .
The big pyramid, with a 36" height,
is times taller,
so its volume is
.
The medium side pyramid above the lower red cut,
being 24" tall is times as tall as the small one,
and had times its volume.
The "stump" piece of the big pyramid below the lower red cut
(it is really called a frustum)
Has a volume of .
The inside pyramid has a height of 12"so its volume is
,
and the part of that pyramid above the green line
Has a height, in inches, of ,
So comparing it to the 12" high small pyramid,
We see that the part above the green line has a volume off
.
So, the volume we have to dig out (another frustum) is
.
After we subtract that volume from
the volume of the first frustum,
the volume of the pot, in cubic inches is
.

NOTE: I cheated a little,
because otherwise the arithmetic would really get ugly.
The pot wall was supposed to be 3" thick.
I gave that measure to the horizontal edge of the pot wall.
Because that is not perpendicular to the walls' surfaces,
the thickness of the walls,
measured perpendicular to the walls surfaces,
as you would measure it with a caliper,
would really be smaller by a factor of .
That is almost .

If you want a second opinion,
you may try the mathalino website.
Those guys enjoy engineering calculations,
and do not mind ugly arithmetic.
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