2014 AMC 10A Problem 19

Question

Four cubes with edge lengths

1,

2,

3,

and

4

are stacked as shown. What is the length of the portion of

\overline{XY}

contained in the cube with edge length

3?

\dfrac{3\sqrt{33}}5

2\sqrt3

\dfrac{2\sqrt{33}}3

4

3\sqrt2

The whole segment

XY

has vertical change

10

The part inside the side-

3

cube is the same fraction of the full segment as its vertical change

The distance between

X

and

Y

with respect to the

z

-axis is

1 + 2 + 3 + 4 = 10.

Both the distances along the

x

and

y

-axes are

4.

Then

XY = \sqrt{4^2 + 4^2 + 10^2} = 2\sqrt{33}.

Using coordinates

X=(0,0,10)

and

Y=(4,4,0)

, the line meets the top and bottom of the side-

3

cube at

(\frac65,\frac65,7)

and

(\frac{12}5,\frac{12}5,4)

. Both points lie inside those square faces, so the portion inside this cube really does have vertical change

3

.

Let the desired length be

x.

Then using similar triangles, we have that

\dfrac{x}{3} = \dfrac{2\sqrt{33}}{10}

x = \dfrac{3\sqrt{33}}{5}.

Thus, A is the correct answer.

Problem 19 in other years