2003 AIME II Problem 5

"A cylindrical log has
a diameter of 12 inches.

A wedge is cut from the log
by making two planar cuts that

go entirely through the log.

The first is perpendicular
to the axis of the cylinder,

and the plane of the second
cut forms a 45 degree angle

with the plane of the first cut.

The intersection
of these two planes

has exactly one point
in common with the log.

The number of cubic
inches in the wedge

can be expressed as n pi,
where n is a positive integer.

Find n."

So let's think about it.

So let's just draw what
they're describing.

So we have a
cylindrical log that

has a diameter of 12 inches.

So let's draw that.

So let me draw it like this.

So that is a cross
section of the log.

It has a diameter of 12 inches.

So this is 12 inches.

And it is a cylinder,
so it looks like this.

That is our log.

That is the log in question.

A wedge is cut from the log
by making two planar cuts that

go entirely through the log.

The first cut is perpendicular
to the axis of the cylinder.

So it would go-- it'd
really just cut it--

it would just cut it straight.

That's another way
to think about it.

It's perpendicular to
the axis of the cylinder.

The axis of the cylinder
goes through the cylinder,

like that.

I don't want to do
that; it'll make

the diagram a little confusing.

If the log was transparent,
our cut would look like this.

This first cut would just go
through the log like this.

It would really just form
a cross section of the log.

So that's the first cut.

The first is perpendicular
to the axis of the cylinder.

And the plane of the second
cut forms a 45 degree angle

with the plane of the first cut.

So that thing is going to
come in at a 45 degree angle.

So it's going to cut into
our log something like that.

Something like that,
right over there.

The intersection
of these two planes

has exactly one point
in common with the log.

That would be this
point, right over here.

So we need to find the
number of cubic inches

in the wedge, expressed
as n pi, and then we

have to figure out n.

So let's draw this wedge, this
wedge that we've cut out here.

So this right here is our wedge.

Let me take it out and
kind of flip it around.

So the base of--
I'll make the base

of the wedge the
perpendicular cut.

This cut right over here.

So that is the
base of the wedge.

That is our base.

And then you can do the top
of our cut to be the magenta,

the 45 degree angle cut.

So maybe I'll draw it like this.

So it would look
something like this.

I'm trying my best to draw it.

That's really the
hard part, here.

So let me draw the
45 degree angle cut.

It's going to look
something like that, where

this angle right over
here-- so if I were

to go-- the diameter of this
top thing, it's actually,

it's not going to
be a normal circle.

It's going to be
more elliptical.

But the diameter
of this top thing

versus the diameter
of the base, which

is the diameter of a
circle, this right here

is going to be a
45 degree angle.

Now when I first
looked at this problem,

there's all sorts
of temptations here.

Maybe you use some calculus.

Maybe you use some-- you rotate
something around some axis

to find the volume.

Maybe you can take some
type of average, here.

And you probably could
do something like that.

But the easiest
thing here-- and this

is kind of-- whenever
you see these kind

of competition-type
math problems--

and this comes from
the 2003 AIM exam--

is that there should be
a quick way to do it.

And for this exam,
particular, you

shouldn't have to use
any type of calculus.

And so, if you find
yourself doing something

tedious and hairy,
you're probably not

seeing the easiest way
to do this problem.

And this problem is actually
ridiculously easy to do,

if you just see the trick here.

And the trick here is,
instead of just directly

trying to solve for the volume
of this figure, right here,

take another one of these guys,
and flip it over, and put it

on top of him.

So let's say, if
you did that, you

would have another thing on
top of it, just like that.

So if I just took
two of these wedges

and stacked them on top of each
other-- flipped one of them

and stacked them on top of each
other, it would look like that.

This is another
wedge right here.

So I took their angled
faces and put them

right on top of each other.

And so if you took two wedges
together, flip one of them,

and put it on top of the
other, what do you get?

So the equivalent-- I could
draw the green wedge right over

here.

The green wedge
would look like this,

where its base looks like this.

So what would two wedges put in
this configuration look like?

Well, it's just a cylinder, now.

And it now is a cylinder.

So this is a cylinder
with a diameter of 12.

So that diameter is 12.

But in order to figure out
the volume of a cylinder,

you still need to figure out
the height of the cylinder.

You still have to figure out
this length, right over here.

You still have to figure out
what the height is equal to.

You have to figure out what
this length, right over here,

is equal to.

And that's where the
45 degrees helps us.

Well, the 45 degrees
already helped us,

because if you flipped the
thing over and put it on top,

it forms a nice cylinder for us.

If it was another
angle, it wouldn't

have been a nice, clean
cylinder, like this.

But the 45 degrees also
tells us what this height is.

And let's just think about
it, here, for a second.

This triangle that I had
drawn, in the beginning--

let me draw this in--
well, I already used blue.

Let me draw it in yellow.

So if I were to
take the diameter

of the angled surface-- and
once again, it's not a circle,

but it's kind of a
stretched-out circle--

and I took the
diameter of the base,

they form a 45 degree angle.

So this is 45 degrees.

This right here is also
going to be 45 degrees.

This length, over
here, we know--

we know this length, right
over here, is going to be 12.

Now, this right here
is a 45-45-90 triangle.

Let me draw it like this.

I could draw it like this.

This right here is
45-45-90 triangle.

And you might say, how
did I know that's 45?

Well, this is going to be a
right angle, right over here.

And the sum of the angles
have to add up to 180.

And then, if you have
45 and 90 already, then

this one has to be 45 degrees.

And in 45-45-90 triangle, this
side is equal to that side.

It is an isosceles triangle.

The two base angles
are the same,

so the two sides are
going to be the same.

So if this side, right
over here, is 12,

then this side, right
over here-- this side,

right over here is 12.

So the height is going
to be equal to 12.

So let's figure out the
volume of this cylinder that's

essentially two
wedges, and then we

can take half of that to
find the volume of one wedge.

And to find the
volume of a cylinder,

you just have to find the area
of the top of the cylinder.

And to find that, it's going to
be pi times the radius squared.

The radius here is 6, 1/2 of 12.

So the area is pi r squared.

So it's 36 pi--
that's that area--

times the height-- times 12.

And so the whole
volume, our volume

is going to be equal
to-- what's that?

360 plus 70.

Let me just multiply it out.

I don't want to make
a careless mistake.

36 times 12.

36 times 2 is 72.

1 times 36 is 36.

2, 13, 4.

So it's 432 pi.

Now we have to be very careful.

This is the volume
of two wedges.

So this is the volume of
two wedges, I could call it.

So the volume of one wedge,
is going to be 1/2 of this.

Let me do this in
a different color.

The volume of one wedge is going
to be 1/2 of this, or 216 pi.

And so if we want to
find n, because they say,

the number of cubic
inches in the wedge

can be expressed as n pi.

It's 216 pi, where n
is a positive integer.

Find n.

Well, we just figured
that out. n is 216.